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Cops & Robber on Periodic Temporal Graphs∗
Jean-Lou De Carufel1, Paola Flocchini1, Nicola Santoro2, and Fr´ed´eric Simard1
1School of Electrical Engineering and Computer Science, University of Ottawa
2School of Computer Science, Carleton University
Abstract
We consider the Cops and Robber pursuit-evasion game when the edge-set of the graph
is allowed to change in time, possibly at every round. Specifically, the game is played on an
infinite periodic sequence G= (G0, . . . , Gp−1)∗of graphs on the same set Vof nvertices: in
round t, the topology of Gis Gi= (V, Ei) where i≡t(mod p).
Concentrating on the case of a single cop, we provide a characterization of copwin periodic
temporal graphs, establishing several basic properties on their nature, and extending to the
temporal domain classical C&R concepts such as covers and corners. Based on these results,
we design an efficient algorithm for determining if a periodic temporal graph is copwin.
We also consider the case of k > 1 cops. By shifting from a representation in terms
of directed graphs to one in terms of directed multi-hypergraphs, we prove that all the
fundamental properties established for k= 1 continue to hold, providing a characterization
of k-copwin periodic graphs, as well as a general strategy to determine if a periodic graph is
k-copwin.
Our results do not rely on any assumption on properties such as connectivity, symmetry,
reflexivity held by the individual graphs in the sequence. They are established for a unified
version of the game that includes the standard games studied in the literature, both for
undirected and directed graphs, and both when the players are fully active and when they
are not. They hold also for a variety of settings not considered in the literature.
∗In occasion of Ralf Klasing’s 60th birthday, celebrating his many outstanding contributions in the area of
discrete algorithms, in particular for mobile agents and dynamic networks, which are the topics of our paper.
1
arXiv:2410.22618v1 [cs.DM] 30 Oct 2024
1 Introduction
1.1 Framework and Background
1.1.1 Cops & Robber Games
Cops & Robber (C&R) is a pursuit-evasion game played in rounds on a finite graph Gbetween
a set of k≥1 cops and a single robber. Before starting the game, an initial position on the
vertices of Gis chosen first by the cops, then by the robber. Then, in each round, first the cops,
then the robber, move to neighbouring vertices or (if allowed by the variant of the game) stay
in the current location. The game ends if the cops capture the robber: at least one cop moves
to the vertex currently occupied by the robber, in which case the cops have won. The robber
wins by forever avoiding capture; note that, in this case, the game never ends.
In the original version, introduced by Quillot [39] and independently by Nowakowski and
Winkler [37], the graph Gis connected and undirected, there is a single cop and, in each round,
the players are allowed not to move; it has then been extended by Aigner and Fromme [2]
to permit multiple cops. This version, which we shall call standard, is the most commonly
investigated (see [5]).
Among the many variants of this game (for a partial list, see [4, 5]), two are of particular
interest to us. The first is the (much less investigated) natural generalization when the graph
Gis a strongly connected directed graph [14, 32, 34]; we shall refer to this version as directed.
Also of interest is the variant, called fully active (or restless), in which the players must move
in every round [13, 26]; proposed for the standard game, this variant can obviously be extended
also to the directed version.
In the extensive existing research (see [5] for a review), the main focus is on characterizing
the class of k-copwin graphs; i.e., those graphs where there exists a strategy allowing kcops to
capture the robber regardless of the latter’s decisions. Related questions are to determine the
minimum number of cops capable of winning in G, called the copnumber of G, or just to decide
whether kcops suffice. The goal underpinning this research is the identification of properties
that allow the characterization of graph classes by means of the copnumber of their members.
The main algorithmic question is the complexity of deciding whether or not a graph is k-
copwin as a function of the input parameters: the number nof vertices, the number mof edges,
and the number kof cops; particular attention has been given to the case when there is a single
cop. Currently, the most efficient algorithm for deciding whether or not a graph is k-copwin in
the standard game is O(knk+2) [38], which yields O(n3) for the case k= 1.
In the existing literature on the C&R game, with only a couple of recent exceptions, all
results are based on a common assumption: the graph on which the game is played is static;
that is, its link structure is the same in every round.
The question naturally arises: what happens if the C&R game is played on a time-varying
graph? More precisely, what happens if the link structure of the graph on which the game is
played changes in time, possibly in every round? In addition to opening a new theoretical line of
inquiry, this question is particularly relevant in view of the intense focus on time-varying graphs
in the last two decades by researchers from several fields.
1.1.2 Temporal Graphs
The extensive investigations on properties and computatonal aspects of time-varying graphs have
been originally motivated by the development and increasing importance of highly dynamic
networks, where the topology is continuously changing. Such systems occur in a variety of
2
different settings, ranging from wireless ad-hoc networks to social networks. Various formal
models have been advanced to describe the dynamics of these networks in terms of the dynamics
of the changes in the graphs representing their topology (e.g., [10, 28, 42]).
When time is discrete, as in the C&R games, the dynamics of these networks is usually
described as an infinite sequence G= (G0, G1, . . . ), called temporal graph (or evolving graph), of
static graphs Gi= (V, Ei) on the same set Vof vertices; the graph Giis called snapshot (of G
at time i), and the aggregate graph G= (V, ∪iEi) is called the footprint (or underlying) graph.
This model, originally suggested in [27] and independently proposed in [21], has become the
de-facto standard in the ensuing investigations.
All the studies are being carried out under some assumptions restricting the arbitrariness
of the changes. Some of these assumptions are on the “connectivity” of the graphs Giin the
sequence; they range from the (strong) 1-interval connectivity requiring every Gito be connected
(e.g., [11, 12, 29, 33]), to the weaker temporal connectivity allowing each Gito be disconnected
but requiring the sequence to be connected over time (e.g., [9, 24]).
Another class of assumptions is on the “frequency” of the existence of the links in the
sequence. An important assumption in this class is periodicity: there exists a positive integer p
such that Gi=Gi+pfor all i∈Z(e.g., [22, 30, 31]). Its importance follows from the fact that
it models a condition occurring in a large variety of important settings, ranging from public
transit networks to low-orbit satellite networks, to activity schedules.
There is a large number of studies on mobile entities operating in temporal graphs, under
different combinations of the above (and other) restrictive assumptions. Among them, compu-
tations include graph exploration,dispersion, and gathering (e.g., [1, 6, 16, 17, 18, 19, 24, 25, 29];
for a recent survey see [15]). Until very recently, none of these studies considered C&R games.
1.2 C&R in Temporal Graphs
Conceptually, the extension of a C&R game to a temporal graph G= (G0, G1, . . . ) is quite
natural. Initially, first the cops, then the robber, choose a starting position on the vertices of
G0. At the beginning of round t≥0, the players are in Gtand, after making their decisions
and moves (according to the rules of the game), they find themselves in Gt+1 in the next round.
The game ends if and only if a cop moves to the vertex currently occupied by the robber; in this
case the cops have won. The robber wins by forever preventing the cops from winning.
1.2.1 Existing Results
The extension of C&R games to temporal graphs has been first investigated by Erlebach and
Spooner [20]. They considered the standard game with a single cop under the periodic frequency
restriction, i.e., the sequence defining Gis periodic, each Giis undirected, players are allowed
not to move, and k= 1. For this setting, they presented an algorithm that determines whether a
periodic temporal graph is copwin in time O(p n3) where pis the period of Gand nthe number
of vertices. They also stated that their algorithm can be extended to k > 1 cops with a resulting
O(k p nk+2) time complexity. In this pioneering study, the results are obtained by reformulating
the problem in terms of a reachability problem and solving the latter; this, unfortunately does
not provide insights on the special nature of the game when the graph changes in time.
Using the same reduction to reachability games, Balev et al. [3] studied the standard game
in temporal graphs under the 1-interval connectivity restriction within a fixed time window,
and indicated how their algorithm can be extended to the case of k > 1 cops. Also in this
case, unfortunately, these results provide no insights on the nature of the game in the temporal
3
dimension. They also considered an “on-line” version of the problem, i.e., where the sequence
of graphs is a priori unknown; these results however are not relevant for the “full-disclosure”
problem studied here.
Finally, if the temporal graph is not given explicitly (i.e., as the sequence of snapshots),
but only implicitly by means of the Boolean edge-presence function1, the problem of deciding
whether a single cop has a winning strategy in the standard game on a periodic temporal graph
has been shown to be N P -hard [35], answering a question raised in [20]. It has been later shown
that the problem is N P -hard even if the footprint G= (V, ∪iEi) of the temporal graph is a very
simple graph (i.e., directed and undirected cycle) [36, 41].
With a different focus, the study of the structural properties of copwin temporal graphs has
just started, examining the relationship between the copnumber of temporal graphs and that of
its static components (e.g., snapshots and footprint) [7, 40].
1.3 Contributions
In this paper we focus on C&R games in periodic temporal graphs, concentrating on the case of
asingle cop.
We study the unified version of the game defined as follow: In every round i≥0, the
snapshot graph Giis directed and the players are restless. Let us point out that the standard
version, both in the original or restless variant, as well as the non-restless directed version can
actually be redefined as a restless game played on (appropriately chosen) directed graphs: a pair
of directed edges between a pair of nodes corresponds to an unidirected link between them, and
the presence of a self-loop at a node allows the players currently there not to move to a different
node in the current round. In other words, the restless directed version of the game includes all
the different versions mentioned above. In this unified version, for the C&R game to be defined,
and thus playable, the only requirement is that every node in the graph must have an outgoing
edge. In our investigation, we will use this simple unified version.
For the unified game, we provide a complete characterization of copwin periodic temporal
graphs, establishing several basic properties on the nature of a copwin game in such graphs. We
do so by using a compact representation of periodic temporal graphs as static directed graphs,
we call arenas, introducing the novel notion of augmented arenas, and using these structures to
extend to the temporal domain classical concepts such as covers and corners.
These characterization results are general, in the sense that they do not rely on any assump-
tion on properties such as connectivity, symmetry, reflexivity held (or not held) by the individual
snapshot graphs in the sequence.
Based on these results, we design an algorithm for determining if a periodic temporal graph
is copwin, prove its correctness and analyze its time complexity. The total cost of the algorithm
is O(p n2+nm), where m=Pi∈Zp|Ei|is the number of edges in the first psnapshots. Thus,
in periodic graphs with sparse snapshots. the proposed algorithm terminates in O(p n2) time,
which, in the static case, becomes O(n2). Following the preliminary announcement of these
results in [8], it has been recently shown that also the reduction to reachability games of [20]
can achieve the same bound [41].
We then consider the case k > 1 of multiple cops. By shifting from a representation in terms
of directed graphs to one in terms of directed multi-hypergraphs, it is possible to extend all
the basic concept introduced for k= 1. Indeed, we prove that all the fundamental properties
of augmented arenas established for k= 1 continue to hold in this extended setting, providing
1The edge-presence function f(e, t)∈ {0,1}indicates for every ein the footprint of Gand round twhether or
not edge eis present in Gt[10].
4
a complete characterization of k-copwin periodic graphs. These results lead directly to a so-
lution strategy to determine if a periodic temporal graph is k-copwin; however, the immediate
implementation of the strategy does not lead to an improved time bound.
All our results are established for the unified version of the game. Therefore, all the charac-
terization properties and algorithmic results hold not only for the standard games studied in the
literature but also for the much less studied directed games, both when the players are restless
and when they are not. They hold also for all those settings, not considered in the literature,
where there is a mix of nodes: those where the players must leave and those where the players
can wait; furthermore such a mix might be time-varying (i.e., different in every round).
2 Terminology and Definitions
2.1 Graphs and Time
2.1.1 Static Graphs
We denote by G= (V, E ), or sometimes by G= (V(G), E(G)), the directed graph with set of
vertices Vand set of edges E⊆V×V. A self-loop is an edge of the form (u, u); if (u, u)∈E
for all u∈V, then we will say that Gis reflexive. If (v, u)∈Ewhenever (u, v)∈E, we will say
that Gis symmetric (or undirected).
Given a vertex u∈V(G), we shall denote by E−(u) the set of edges incident on u, and by
and by E+(u) those departing from u. A vertex v∈V(G) is said to be a source if E−(u) = ∅,
and to be a sink if E+(u) = ∅.Gis said to be sourceless if it contains no sources, and sinkless
if it contains no sinks.
Given a graph G′, if V(G′)⊆V(G) and E(G′)⊆E(G), then we say G′is a subgraph of G
and write G′⊆G. A subgraph G′⊆Gis proper, written G′⊂G, if G′=G.
For reasons apparent later, we shall refer to a graph Gso defined as a static graph, and say
it is playable if every vertex has at least one outgoing edge.
2.1.2 Temporal Graphs
Atime-varying graph Gis a graph whose set of edges changes in time2. A temporal graph is a
time-varying graph where time is assumed to be discrete and to have a start, i.e., time is the set
Z+of positive integers including 0.
A temporal graph Gis represented as an infinite sequence G= (G0, G1, . . . ) of static graphs
Gi= (V, Ei) on the same set of vertices V; we shall denote by n=|V|the number of vertices. The
graph Giis called the snapshot of Gat time i∈Z+, and the aggregrate graph G= (V , SiEi)
is called the footprint of G. A temporal graph Gis said to be reflexive if all its snapshots
are reflexive, symmetric if all its snapshots are symmetric, sourceless if all its snapshots are
sourceless.
Given two verices x, y ∈V, a strict journey (or temporal walk), from xto ystarting at
time tis any finite sequence π(x, y) = ⟨(z0, z1),(z1, z2),...,(zk−1, zk)⟩where z0=x, zk=y,
and (zi, zi+1)∈Et+ifor 0 ≤i<k. In the following, for simplicity, we will omit the adjective
“strict”.
A temporal graph Gis temporally connected if for any u, v ∈Vand any time t∈Z+there
is a journey from uto vthat starts at time t. Observe that, if Gis temporally connected, then
its footprint is strongly connected even when all its snapshots are disconnected. A temporal
2The terminology in this section is mainly from [10].
5
graph Gis said to be always connected (or 1-interval connected) if all its snapshots are strongly
connected.
A temporal graph Gis periodic if there exists a positive integer psuch that for all i∈Z+,
Gi=Gi+p. If pis the smallest such integer, then pis called the period of Gand Gis said to
be p-periodic. We shall represent a p-periodic temporal graph Gas G= (G0, . . . , Gp−1)∗; all
operations on the indices will be taken modulo p. An example of a temporal periodic graph G
with p= 4 is shown in Figure 1; observe that Gis temporally connected, however most of its
snapshots are disconnected digraphs, and none of them is strongly connected.
Let G= (G0. . . Gp−1)∗and H= (H0. . . Hp−1)∗be two temporal periodic graphs with the
same period on the same set Vof vertices; we say His a periodic subgraph of G, written H ⊆ G,
if Hi⊆Gifor every i∈Zp={0,1, . . . , p −1}. We shall denote by H ⊂ G the fact that His a
proper subgraph of G, i.e., H ⊆ G but H =G. Let us point out the obvious but useful fact that
static graphs are temporal periodic graphs with period p= 1.
In this paper we focus on C&R games in periodic temporal graphs, henceforth referred to
simply as periodic graphs, concentrating on the case of a single cop, and then focusing on the
case of multiple cops.
2.1.3 Arena
Consider the following class of directed static graphs, we shall call arenas.
Definition 2.1 (Arena).Let k≥1 be an integer and Wbe a non-empty finite set. An
arena of lenght kon Wis any static directed graph M= (Zk×W, E(M)) where E(M)⊆
{((i, w),([i+ 1]k, w′))|i∈Zkand w, w′∈W}, and [i]kdenotes imodulo k.
A periodic graph G= (G0, . . . , Gp−1)∗with period pand set of vertices Vhas a unique
correspondence with the arena D= (Zp×V, E(D)) where, for all i∈Zp,
((i, u),([i+ 1]p, v)) ∈E(D)⇐⇒ (u, v)∈Ei,
called the arena of G. In particular, the arena Dof Gexplicitly preserves the snapshot structure
of G: for all i∈Zp, there is an obvious one-to-one correspondence between the snapshot Gi
of Gand the subgraph Siof D, called slice (or stage), where V(Si) = {(i, v), v ∈V}and
E(Si) = {((i, u),([i+ 1]p, v))|(u, v)∈Ei)}. An example of a periodic graph Gand its arena Dis
shown in Figure 1. In the following, when no ambiguity arises, Dshall indicate the arena of G.
The vertices of an arena Dwill be called temporal nodes. Given a temporal node (i, u)∈
V(Si) we shall denote by Ni(u, D) the set of its outneighbours, and by Γi(u, D) = {v∈V|([i+
1]p, v)∈Ni(u, D)}the corresponding set of vertices in Gi.
A temporal node (i, u)∈V(Si) is said to be a star if Γi(u, D) = V. It is said to be anchored
if there exists a journey from some remporal node (0, v)∈V(S0) to (i, u).
Asubarena of D= (Zp×V, E (D)) is any arena D′= (Zp×V, E(D′)) where E(D′)⊆E(D);
we shall denote by D′⊂ D the fact that D′is a subarena of Dwith E(D′)⊂E(D).
2.2 Cop & Robber Game in Periodic Graphs
2.2.1 Basics
The extension of the game from static to temporal graphs is quite natural. Initially, first the
cop, then the robber, chooses a starting position on the vertices of G0. Then, at each time
t∈Z+, first the cop, then the robber, moves to a vertex adjacent to its current position in Gi,
6
G1
G2
G3
d
c
G0
a
b
c
d
t= 0
t= 1
t= 2
t= 3
t= 0
S0
S1
S2
S3
a
b
G
a
b
a
b
a
b
a
b
d
c
d
c
d
c
d
c
Figure 1: A periodic graph G= (G0, G1, G2, G3)∗, its footprint G, and the corresponding arena.
where i= [t]p. Thus, in round t, the players are in G[t]pand, after making their decisions and
moves, they find themselves in G[t+1]pin the next round. The game ends if and only if the cop
moves to the vertex currently occupied by the robber; in this case the cop has won. The robber
wins by forever preventing the cop from winning.
We consider the version of the game where all players are restless, i.e., they all move in each
round. The only requirement made by this version on Gis that it is playable: in each snapshot,
every vertex must have an outgoing edge; that is, every Giis sinkless. In the following we only
consider playable periodic graphs. No other requirement such as connectivity, symmetry, and
reflexivity is imposed on G.
We call this version of the game unified. Observe that the standard version, both in the
original or restless variant, as well as the non-restless directed version can actually be redefined
as a restless game played in this unified version: a pair of directed edges between a pair of verices
corresponds to an unidirected link between them, and the presence of a self-loop at a vertex
allows the players currently there not to move to a different vertex in the current round.
A play on the arena Dof Gfollows the play on Gin a direct obvious way: at each time
t∈Z+, first the cop, then the robber, chooses a new vertex in the out-neighbourhood of its
current position and moves there. The cop wins and the game ends if it manages to move to a
temporal node ([t+ 1]p, u) while the robber is on ([t]p, u). The robber wins by forever escaping
capture from the cop, in which case the game never ends.
2.2.2 Configurations and Strategies
Aconfiguration is a triple (t, c, r)∈Z+×V×V, denoting the position c∈Vof the cop and
r∈Vof the robber at the beginning of round t∈Z+. Let CG = (V(CG), E(CG)) be the infinite
directed graph, called configuration graph of D, describing all the possible configurations (t, u, v)
and their temporal connection in D:
V(CG) = {(t, u, v)|t∈Z+; ([t]p, u),([t]p, v)∈V(D)},
E(CG) = {((t, u, v),(t+ 1, u′, v′))|t∈Z+;u=v;u′∈Γ[t]p(u, D), v′∈Γ[t]p(v, D)}.
7
Observe that CG is acyclic; the source nodes (i.e., the nodes with no in-edges) are those with
t= 0, the sink nodes (i.e., the nodes with no out-edges) are those with u=v.
A playing strategy for the cop is any function σc:V(CG)→Vwhere, for every (t, u, v)∈
V(CG), σc(t, u, v)∈Γ[t]p(u, D), and σc(t, u, v) = uif u=v; it specifies where the cop should
move in round tif the cop is at ([t]p, u), the robber is at ([t]p, v), and it is the cop’s turn to
move. A playing strategy σrfor the robber is defined in a similar way.
A configuration (t, u, v) is said to be copwin if there exists a strategy σcsuch that, starting
from (t, u, v), the cop wins the game regardless of the strategy σrof the robber; such a strategy
σcwill be said to be copwin for (t, u, v). A strategy σcis said to be copwin if there exists a
temporal node (0, u) such that σcis winning for (0, u, v) for all v∈V.
If a copwin strategy exists, then Gand its arena Dare said to be copwin, else they are
robberwin.
3 Copwin Periodic Graphs
3.1 Preliminary
In the analysis of the standard game played in a static graph, an important role is played by
the notions of corner node and its cover. The usual meaning is that if the robber is on the
corner, after the cop has moved to the cover, no matter where the robber plays, the robber gets
captured by the cop in the next round.
In an arena D, the same meaning is provided directly by the notions of “temporal corner”
and “temporal cover”.
Definition 3.1. (Temporal Corner and Temporal Cover) A temporal node (t, u) in an arena D
is said to be a temporal corner of temporal node (t+ 1, v) if u=vand
Γt(u, D)⊆Γt+1 (v, D).
The temporal node (t+ 1, v) is said to be a temporal cover of (t, u).
An important relationship between temporal corners and copwin arenas is the following.
Lemma 3.1. Every copwin arena contains a temporal corner.
Proof. Let Dbe a copwin arena. Then there must exists a time t, a configuration (t, c, r),
and a move by the cop to a neighbouring temporal node (t+ 1, c′) such that, regardless of
where the robber moves in round t, it is captured by the cop in its next move. In other words,
for every w∈Γt(r, D), there exists a z∈Γt+1 (c′,D) such that z=w. This means that
Γt(r, D)⊆Γt+1 (c′,D); that is, (t, r) is a temporal corner of (t+ 1, c′). 2
This necessary condition, although important, provides only limited indications on how to
solve the characterization problem.
3.2 Augmented Arenas and Characterization
The crucial element in the characterization of copwin periodic graphs is the notion of augmented
arena.
8
Definition 3.2. (Augmented Arena) Let Dbe the arena of G. An augmented arena Aof Dis
an arena such that D ⊆ A and, for each edge ((t, x),(t+1, y)) ∈E(A), the configuration (t, x, y)
is winning for the cop in D.
We shall refer to the edges of the augmented arena Aof Das shadow edges. Observe that,
by definition, all edges of Dare shadow edges of A.
Let A(D) denote the set of augmented arenas of D. Observe that, by definition, D ∈ A(D).
Further observe the following:
Property 3.1. The partial order (A(D),⊂)induced by edge-set inclusion on A(D)is a complete
lattice. Hence (A(D),⊂)has a maximum which we denote by A∗.
Proof. It follows from the fact that, by definition of augmented arena, the set A(D) is closed
under the union of augmented arenas. 2
We have now the elements for the characterization of copwin periodic graphs.
Theorem 3.1. (Characterization Property)
An arena Dis copwin if and only A∗contains an anchored star.
Proof. (if) Let A∗contain an anchored star (t, u), t∈Zp. By definition of star, Γt(u, A∗) = V.
Thus, by definition of augmented arena, for every v∈Vthe configuration (t, u, v) is copwin,
i.e., there is a copwin strategy σcfrom (t, u, v).
Since (t, u) is anchored, there exists a temporal node (0, x) such that there is a journey π
from (0, x) to (t, u). Consider now the cop strategy σ′
cof: (1) initially positioning itself on the
temporal node (0, x), (2) then moving according to the journey π((0, x),(t, u)) and, once on
(t, u), (3) following the copwin strategy σcfrom (t, u, w), where wis the position of the robber
at the beginning of round t. This strategy σ′
cis winning for (0, x, v) for all v∈V; hence Dis
copwin.
(only if) Let Dbe copwin. We show that there must exist an augmented arena Aof D
that contains an anchored star. Since Dis copwin, by definition, there must exist some starting
position (0, x) for the cop such that, for all positions initially chosen by the robber, the cop
eventually captures the robber. In other words, all the configurations (0, x, v) with v∈Vare
copwin; thus the arena Aobtained by adding to E(D) the set of edges {((0, x),(1, v))|v∈V}is
an augmented arena of Dand (0, x) is an anchored star. By Property 3.1, E(A)⊆E(A∗) and
the theorem follows. 2
The characterization of copwin periodic graphs provided by Theorem 3.1 indicates that, to
determine whether or not an arena Dis copwin, it suffices to check whether A∗contains a star.
To be able to transform this fact into an effective solution procedure, some additional con-
cepts need to be introduced and properties established.
3.3 Shadow Corners and Augmentation
Other crucial elements in the analysis of copwin periodic graphs are the concepts of corner and
cover, introduced in Section 3.1 for arenas, now in the context of augmented arenas.
Definition 3.3. (Shadow Corner and Shadow Cover) Let Abe an augmented arena of D. A
temporal node (t, u) is a shadow corner of a temporal node (t+ 1, v), with v=u, if
Γt(u, D)⊆Γt+1 (v, A).
The temporal node (t+ 1, v) will then be called the shadow cover of (t, u).
9
u=v
Γt(u, D)⊆Γt+1(v, A)
(t, u) is a shadow corner of (t+ 1, v).
(t+ 1, v) is a shadow cover of (t, u).
t
t+ 1
t+ 2
u
v
Plain and dashed edges are in D.
Dotted edges are in E(A)\E(D).
v
v
u
u
Figure 2: Temporal node (t, u) is a shadow corner of (t+ 1, v).
By definition, any temporal corner is a shadow corner, and its temporal covers are shadow
covers. An example is shown in Figure 2; the dashed links indicate the neighbours of node (t, u)
in D, while the dotted links indicate the edges to the neighbours of (t+ 1, v) that exists in A
but not in D.
The role that shadow corners play with regards to the set A(D) of augmented arena of Dis
expressed by the following.
Theorem 3.2. (Augmentation Property)
Let A ∈ A(D),(t, x),(t, y)∈V(D)and z∈Γt(x, D). If (t, y)is a shadow corner of (t+ 1, z),
then the arena A′=A∪{((t, x),(t+ 1, y))}is an augmented arena of D.
Proof. Let Abe an augmented arena of Dand let (t, x),(t, y),(t+ 1, z)∈V(D) where
z∈Γt(x, D) and (t, y) is a shadow corner of (t+ 1, z ). The theorem follows if ((t, x),(t+1, y)) is
already an edge of A. Consider the case where ((t, x),(t+ 1, y)) /∈E(A). Since (t, y) is a shadow
corner of (t+ 1, z), then for every w∈Γt(y, D) we have that ((t+ 1, z),(t+ 2, w)) ∈E(A), i.e.,
(t+ 1, z, w) is winning for the cop. Since z∈Γt(x, D), if the cop moves from (t, x) to (t+ 1, z)
when the robber is on (t, y), then regardless of the robber’s move, the resulting configuration
would be winning for the cop. In other words, (t, x, y) is a winning configuration for the cop. It
follows that A′=A∪{((t, x),(t+ 1, y))}is an augmented arena of D.2
In other words, given an augmented arena, by identifying a (still unconsidered) shadow corner
and its covers, new shadow edges may be determined and added to form a denser augmented
arena.
3.4 Determining A∗
The properties expressed by Theorem 3.2, in conjunction with that of Theorem 3.1, provide an
algorithmic strategy to construct A∗: start from an augmented arena; determine new shadow
edges; add them to the set of shadow edges, creating a denser augmented arena; repeat this
process until the current augmented arena Aeither contains an anchored star or is A∗.
To be able to employ the above strategy, a condition is needed to determine if the current
augmented arena of Dis indeed A∗. This is provided by the following.
10
(x, y)
t
t+ 1
t+ 2
t+ 3
sink
σc
winning strategy for (t, x, y) (every path terminates)
Figure 3: The directed acyclic graph Cof configurations induced by σcstarting from (t, x, y).
Theorem 3.3. (Maximality Property)
Let A ∈ A(D). Then A=A∗if and only if, for every edge ((t, x),(t+ 1, y)) /∈E(A), there exists
no z∈Γt(x, D)such that (t, y)is a shadow corner of (t+ 1, z).
Proof. (only if ) By contradiction, let A=A∗but there exists an edge ((t, x),(t+1, y)) /∈E(A)
and a temporal node z∈Γt(x, D) such that (t, y) is a shadow corner of (t+1, z). By Theorem 3.2,
A′=A∪{((t, x),(t+ 1, y ))}is an augmented arena of D; however, E(A′) contains one more
edge than E(A), contradicting the assumption that Ais maximum.
(if) Let A =A∗; that is, there exists ((t, x),(t+ 1, y)) ∈E(A∗)\E(A). By definition, the
configuration (t, x, y) is copwin; let σcbe a copwin strategy for the configuration (t, x, y), i.e.,
starting from (t, x, y), the cop wins the game regardless of the strategy σrof the robber.
Let C= (V(C), E(C)) ⊆ CG be the directed acyclic graph of configurations induced by
σcstarting from (t, x, y), and defined as follows: (1) (t, x, y)∈V(C); (2) if (t′, u, v)∈V(C)
with t′≥tand u=v, then, for all w∈Γt′(v, D), (t′+ 1, σc(t′+ 1, u, v), w)∈V(C) and
((t′, u, v),(t′+ 1, σc(t′+ 1, u, v), w))) ∈E(C).
Observe that in Cthere is only one source (or root) node, (t, x, y), and every (t′, w, w)∈V(C)
is a sink (or terminal) node. Since σcis a winning strategy for the root, every node in Cis a
copwin configuration, and every path from the root terminates in a sink node. See Figure 3.
Partition V(C) into two sets, Uand Wwhere U={(i, u, v)|((i, u),(i+ 1, v)) ∈E(A)}
and W=V(C)\U. Observe that every sink of V(C) belongs to U; on the other hand, since
((t, x),(t+ 1, y)) /∈E(A) by assumption, the root belongs to W(see Figure 4).
Given a node κ= (i, u, v)∈V(C), let C[κ] denote the subgraph of Crooted in κ.
Claim. There exists κ∈V(C)such that all nodes of C[κ], except κ, belong to U.
Proof of Claim. Let P0be the set of sinks of C. Starting from k= 0, consider the set Pk+1 of
all in-neighbours of any node of Pk; if Pk+1 does not contains an element of W, then increase
kand repeat the process. Since (t, x, y )∈W, this process terminates for some k≥0, and the
Claim holds for every κ∈Pk+1 ∩W.2
11
W: Shadow edges are missing.
U: Shadow edges are not missing.
Figure 4: The sets W(□) and U(•).
Let (t′, x′, y′) be a node of V(C) satisfying the above Claim (see Figure 5a). Thus ((t′, x′),(t′+
1, y′)) /∈E(A) but, since (t′, x′, y′) is copwin, ((t′, x′),(t′+ 1, y′)) ∈ A∗. By the Claim, all other
nodes of C[(t′, x′, y′)] belong to U, in particular the set of nodes {(t′+1, w, z )|w=σc(t′, x′, y′), z ∈
Γt′(y′,D)}. This means that, for every z∈Γt′(y′,D), (t′+ 1, w, z)∈E(A). In other words,
Γt′(y′,D)⊆Γt′+1 (w, A); that is, (t′, y′) is a shadow corner of (t′+ 1, w) (see Figure 5b).
(x′, y′)
(w, z1)
(w, z2)
(w, zi)
(w, zk)
t′
No shadow edges missing
(a) The situation in C. (t′, x′, y′) satis-
fies the Claim.
t′
t′+ 1
t′+ 2
x′
y′
w
z1
z2
zk
missing
(b) The situation in A. (t′, y′) is a
shadow corner of (t′+ 1, w).
Figure 5: The situation in Cand in A.
Summarizing: by assumption A =A∗; as shown, ((t′, x′),(t′+ 1, y′)) ∈E(A∗)\E(A), and
w∈Γt′(x′,D) is a shadow cover of (t′, y′); that is, Γt′(y′,D)⊆Γt′+1 (w, A), concluding the
proof of the if part of the theorem. 2
4 Algorithmic Determination
In this section we show that the results established in the previous sections provide all the tools
necessary to design an algorithm to determine whether or not a periodic graph Gis copwin.
Furthermore, if Gis copwin, the algorithm can actually provide a winning cop strategy σc.
12
General Strategy
1. While there is a still unexamined shadow edge e= ((t, x),(t+ 1, y)) in Ado:
2. If there are still unexamined shadow corners covered by (t, x) then:
3. For each such shadow corner (t−1, z) do:
4. If there are new shadow edges due to (t−1, z) then:
5. Add them to Ato be examined.
6. Remove (t−1, z) from consideration as a shadow corners of (t, x) (i.e., mark it as examined).
7. Remove efrom consideration (i.e., mark it as examined).
8. If there is an anchored star in A, then Dis copwin else it is robberwin.
Figure 6: Outline of general strategy where the iterative process terminates when A=A∗.
4.1 Solution Algorithm
4.1.1 General Strategy
Given a periodic graph G, or equivalently its arena D, to determine whether or not it is copwin,
by Theorem 3.1, it is sufficient to determine whether or not its maximal augmented arena A∗
contains an anchored star. Hence, informally, a basic solution approach is to start from A=D,
repeatedly determine a “new” shadow edge (i.e., in E(A∗)\E(A)) using Theorem 3.2, and
consider the new augmented arena obtained by adding such an edge. This process is repeated
until either the current augmented arena Acontains an anchored star, or no other “missing”
shadow edge exists. In the former case, by Theorem 3.1, Dis copwin; in the latter case, by
Theorem 3.3, the current augmented arena is A∗and, if it does not contain an anchored star,
Dis robberwin.
A general strategy based on this approach operates in a sequence of iterations, each composed
of two operations: the examination of a shadow edge, and the examination of new shadow corners
(if any) determined in the first operation. More precisely, in each iteration: (i) A “new” (i.e.,
not yet examined) shadow edge e= ((t, x),(t+ 1, y)) is examined to determine if its presence
transforms some temporal nodes into new shadow corners of (t, x). (ii) Each of these new shadow
corners is examined, determining if its presence generates new shadow edges. By the end of the
iteration, the shadow edge eand the new shadow corners of (t, x) examined in this iteration
are removed from consideration. This iterative process continues until there are no new shadow
edges to be examined (i.e. A=A∗) or there is an anchored star in A.
An outline of the strategy, where the iterative process is made to terminate when A=A∗,
is shown in Figure 6.
4.1.2 Algorithm Description
Let us present the proposed algorithm, CopRobberPeriodic, which follows directly the general
strategy described above to determine whether or not an arena D= ((Zp×V), E(D)) is copwin,
where V={v1, . . . , vn}.
We denote by Athe current augmented arena of D, by Aits adjacency matrix, and by At
the adjacency matrix of slice Stof A. Auxiliary structures used by the algorithm include the
queue SE, of the known shadow edges that have not been examined yet; a n×nBoolean matrix
SEtfor each t, initialized to At, used to indicate shadow edges already known; a n×nBoolean
matrix SCtfor each t, initialized to zero and used to indicate the detected shadow corners;
13
more precisely, Bt[x, y] = 1 indicates that (t, x) has been determined to be a shadow corner of
(t+ 1, y),
The algorithm is composed of two phases: Initialization, in which all the necessary structures
are set up and preliminary computations are performed; and Iteration, a repetitive process where
the two basic operations of the general strategy (described in Section 4.1.1) are performed in each
iteration: examination of a “new” shadow edge (to determine “new” shadow corners generated
by that edge) and examination of the “new” shadow corners (to determine “new” shadow edges
generated by that corner).
The structure used to determine new shadow corners is the set {DIF(t, x, y) : t∈Zp, x, y ∈
V}of n2pBoolean arrays of dimension n. For all x, v ∈Vand t∈Zp, the value of the cell
DIF(t, x, y)[i] indicates whether
vi∈Γt(x, D)\Γt+1 (y, A)
(in which case DIF(t, x, y)[i] = 1) or
vi∈Γt(x, D)∩Γt+1 (y, A)
(in which case DIF(t, x, y)[i] = 0). Note that, if vi/∈Γt(x, D), the value of DIF(t, x, y)[i] is left
undefined; indeed, the algorithm only initializes and uses the |Γt(x, D)|cells corresponding to
the elements of Γt(x, D); we shall call those cells the core of DIF(t, x, y).
The algorithm also maintains a variable ϕ(DIF(t, x, y)) indicating the current number of
core cells with value “1” in array DIF(t, x, y); this variable is initialized to |Γt(x, D)|. Observe
that, by definition of DIF(t, x, y), ϕ(DIF(t, x, y)) = 0 iff (t, x) is a shadow corner of (t+ 1, y).
In each iteration of the Iteration phase, a new shadow edge is taken from SE, added to the
augmented arena A, and examined. The examination of a shadow edge ((t, x),(t+1, y)) involves
(i) the update of DIF(t−1, z, x)[y] for any in-neighbour (t−1, z), in D, of (t, y) and, for any such
in-neighbour, (ii) the test to see if the presence of the edge ((t, x),(t+ 1, y)) in the augmented
arena has created new shadow corners among such in-neighbours3. If new shadow corners exist,
they may in turn have created new shadow edges originating from the in-neighbours, in D, of
(t, x). In fact, any in-neighbour (t−1, w) of (t, x) such that ((t−1, w),(t, z)) is not already in
the augmented arena is a new shadow edge: a move of the cop from (t−1, w) to (t, x) is fatal
for the robber wherever it goes; in such a case, the algorithm then adds ((t−1, w),(t, z)) to SE.
The pseudo code of the algorithm is shown in Algorithm 1. Observe that, in the algorithm,
the set of in-neighbours of a temporal node (t, v) is denoted by Γin
t(v, D) = {z∈V|((t−
1, z),(t, v)) ∈E(D)},
Not shown are several very low level (rather trivial) implementation details. These include,
for example, the fact that the core cells of DIF(t, x, y) are connected through a doubly linked list,
and that, for efficiency reasons, we also maintain two additional doubly linked lists: one going
through the core cells of the array containing “1”, the other linking the core cells containing
“0”.
3Such would be any (t−1, z) for which the update has resulted in an array DIF(t−1, z, x) that contains only
zero entries.
14
Algorithm 1: CopRobberPeriodic
Input: Arena D= (Zp×V, E (D)), with V={v1, . . . , vn}
1Initialization
2A:= D
3SE := A
4SE =∅
5SC :=Zero /* a table of pzero matrices, each of size n×n*/
6foreach t∈Zp,u, v ∈Vdo
7ϕ(DIF(t, u, v)) := |Γt(u, D)|
8foreach w∈Γt(u, D)do
9if At+1[v, w]=1then
10 DIF(t, u, v)[w] := 0
11 ϕ(DIF(t, u, v)) := ϕ(DIF(t, u, v)) −1
12 if ϕ(DIF(t, u, v)) = 0 and SCt[u, v]=0then
13 SCt[u, v] := 1
14 foreach z∈Γin
t+1 (v, D)do
15 if SEt[z , u]=0then
16 SEt[z , u] := 1
17 SE ← ((t, z),(t+ 1, u))
18 else
19 DIF(t, u, v)[w] := 1
20 Iteration
21 while SE =∅do
22 ((t, x),(t+ 1, y)) ← SE
23 At(x, y) := 1
24 foreach z∈Γin
t(y, D)do
25 if DIF(t−1, z, x)[y] = 1 then
26 DIF(t−1, z, x)[y] := 0
27 ϕ(DIF(t−1, z, x)) := ϕ(DIF(t−1, z, x)) −1
28 if ϕ(DIF(t−1, z, x)) = 0 and S Ct−1[z, x] = 0 then
29 SCt−1[z , x] := 1
30 foreach w∈Γin
t(x, D)do
31 if SEt−1[w, z]=0then
32 SEt−1[w, z] := 1
33 SE ← ((t−1, w),(t, z ))
34 if Acontains an anchored star then Dis copwin
35 else Dis robberwin.
15
4.2 Analysis
4.2.1 Correctness
Let us prove the correctness of Algorithm CopRobberPeriodic. Let D= (Zp×V, E (D)) be
the arena of a p-periodic graph with n=|V|and m=|E(D)|.
Lemma 4.1. Algorithm CopRobberPeriodic terminates after at most |E(A∗)|−|E(D)|it-
erations.
Proof. In the Initialization phase, all of the m=|E(D)|edges of Dare examined, and their
entry in the shadow edge matrix SE is set to 1 (Line 3). Any new shadow edge discovered in
this phase is inserted in SE (Line 17).
Observe that, when a shadow edge eis inserted in SE, its entry in the shadow edge matrix
SE is set to 1 (this is done in Line 16 for the edges of D, and in Line 32 for the others); this
means that, once extracted and examined, ewill fail the test of Line 15 (or the test of Line 31)
in any subsequent iteration and, therefore, it will never be inserted in SE again. Since only one
shadow edge is extracted from SE and examined in each iteration, the number of iterations is
at most the total number |E(A∗)|−|E(D)|of shadow edges not originally in D.2
Given an augmented arena Aand a shadow edge e= ((t, x),(t+ 1, y)) ∈E(A∗)\E(A), we
shall say that eis an implicit shadow edge of Aif there exists z∈Γt(x, D) such that (t, y) is a
shadow corner of (t+ 1, z) in A.
Lemma 4.2. At the end of the Initialization phase: (i)for all and only the temporal corners
(t, x)of (t+ 1, y)in D,SCt[x, y]=1and ϕ(DIF(t, x, y)) = 0;(ii)all implicit shadow edges of
Dare in SE; furthermore, the entry in SE of all edges of Dand implicit shadow edges of D, is
1.
Proof.
(i) Observe that, in the Initialization phase, by construction, ∀t∈Zp,∀(t, u),(t+ 1, v)∈
V(D), and ∀w∈Γt(u, D), DIF(t, u, v) is initialized so that DIF(t, u, v)[w] = 1 if and only if
w∈Γt(u, D)\Γt+1 (v, D). Every time it is determined that DIF(t, u, v)[w] = 0, the counter
ϕ(DIF(t, u, v)), initialized to |Γt(u, D)|, is decreased by one; thus, by definition, ϕ(DIF(t, u, v)) =
0 if and only if (t, u) is a temporal corner of (t+ 1, v); in such a case SCt[u, v] = 1. Recall that,
by definition, all temporal corners are also shadow corners.
(ii) First observe that all edges of Dare by definition shadow edges, and that their corre-
sponding entry in the shadow edges matrix SE is set to 1 (Line 3); hence, for them, the lemma
holds.
Let us now consider the implicit shadow edges of D. Recall that, by Theorem 3.2, given a
shadow corner (t, u) of (t+1, v), any edge originating from an in-neighbour (t, z) of (t+1, v) and
terminating in (t+ 1, u) is a shadow edge (Lines 14-17); hence, it is immediate to identify the
implicit shadow edges corresponding to a given shadow corner. An implicit shadow edge (i.e.,
one whose entry in SE is 0), once identified, is added to the queue SE, and the corresponding
entry in the shadow edges matrix is set to 1. Since, by part (i) of this Lemma, all the shadow
corners present in Dare identified, it follows that all the implicit shadow edges are queued in
SE and their entry in SE is set to 1. 2
Let us consider the Initialization phase as iteration 0 of the Iteration phase; hence, the entire
algorithm can be viewed as a sequence of iterations. Denote by Ajthe augmented arena at the
beginning of the j-th iteration, with A0=D. We now show that, at the beginning of iteration
16
j, all shadow corners of Aj−1have been examined and all implicit shadow edges of Aj−1are in
SE.
Lemma 4.3. At the beginning of iteration j > 0:
(a) ϕ(DIF(t, x, y)) = 0 if and only if (t, x)is a shadow corner of (t+1, y)in Aj−1; furthermore,
in such a case, SCt[x, y]=1.
(b) SE contains all the implicit shadow edges of Aj−1; furthermore, in SE, the entry of the
edges of Aj−1and of the implicit shadow edges of Aj−1is 1.
Proof. By induction on j. Observe that, when j= 1, both statements of the lemma follow
directly from Lemma 4.2. Let they hold for j≥1; we now prove that they hold for j+ 1.
Let ej= ((t, x),(t+ 1, y)) be the shadow edge extracted from SE and examined in iteration
j. This edge is added to Aj−1(Line 23), which is thus transformed into Aj. This addition,
which modifies only the out-neighbourhood of (t, x), might create new shadow corners of (t, x)
among the in-neighbours of (t, y). The algorithm therefore checks the set Γin
t(y, D) to verify if
this has happened (Lines 24-33). This is done by considering, for each element (t−1, z) of that
set, the entry DIF(t−1, z, x)[y].
If DIF(t−1, z, x)[y] = 1, then the shadow edge ejwas one of those missing edges; hence,
in that case (Lines 25-27) the value of DIF(t−1, z, x)[y] is set to zero and ϕ(DIF(t−1, z, x))
is decreased by one. If now ϕ(DIF(t−1, z, x)) becomes 0, then (t−1, z ) is a shadow corner of
(t, x) in Ajand SCt−1[z , x] is set to 1 (Line 29). This means that, at the end of this iteration,
all shadow corners of Aj\ Aj−1have their entry in SC set to 1 and the corresponding entry in
ϕ(·) set to 0; thus, by the inductive hypothesis on the shadow corners of Aj−1, statement (a)
of the lemma holds for iteration j.
Any new shadow corner (t−1, z) of (t, x), created by the addition of ej, might in turn have
created new implicit shadow edges in Aj. By Theorem 3.2, any edge originating from an in-
neighbour (t−1, w) of (t, x) and terminating in the shadow corner (t, z) is a shadow edge (Lines
30-33). Let Pdenote the set of these shadow edges; among them, the only implicit ones for
Ajare, by inductive hypothesis, the ones whose entry in SE was 0 in Aj−1. Any such implicit
shadow edge is thus identified, added to the queue SE, and the corresponding entry in SE is
set to 1. Since, by part (a) of this Lemma, all the shadow corners present in Ajare identified,
it follows that all the new implicit shadow edges of Ajare queued in SE and their entry in SE
is set to 1. Thus, by inductive hypothesis on the shadow edges of Aj−1, statement (b) of the
lemma holds for iteration j.2
Theorem 4.1. Algorithm CopRobberPeriodic correctly determines whether or not an arena
Dis copwin.
Proof. By Lemma 4.1, the algorith terminates after a finite number q≥1 of iterations,
when SE becomes empty and no other shadow edges are added to it during the iteration. By
Lemma 4.3, the fact that SE =∅at the end of the iteration means that in Aqthere are no implicit
shadow corners identified in previous iterations; furthermore, during this iteration, regardless of
the new shadow corners found and examined, no implicit shadow corners were found. In other
words, the set E(A∗)−E(Aq) = ∅, i.e., Aq=A∗. Hence, by Theorem 3.3, the test in the last
operation of the algorithm (Lines 34-35) determines correctly whether or not Dis copwin. 2
17
4.2.2 Complexity
Let us analyze the cost of Algorithm CopRobberPeriodic. Given D= (Zp×V, E(D)), let
midenote the number of edges of slice Siof D,i∈Zp, and m=|E(D)|=Pp−1
i=0 mithe total
number of edges of D. As usual, n=|V|.
Theorem 4.2. Algorithm CopRobberPeriodic determines in time O(n2p+nm)whether or
not Dis copwin.
Proof. We first derive the cost of the Initialization phase. Observe that the initialization of
A,SE,SC (Lines 2-4) can be performed with O(n2p) operations. Line 7 will be executed n2p
times. The cost of the initialization of DIF and of ϕ(DIF) (Lines 6-13,18-19), which includes
the update of some entries of SC , plus the cost of the initialization of S E (Lines 14-17), which
includes the update of some entries of SE, require at most
O(n2p) + X
i∈Zp,u∈V
O(|Γi(u, D)|) + X
i∈Zp,v∈V
O(|Γin
i(v, D)|) = O(n2p) +
p−1
X
i=0
O(n(mi+mi−1))
operations, which sums up to O(n2p+nm) operations for the Initialization phase.
Let us consider now the Iteration phase. The while loop will be repeated until in the current
augmented arena Athere are no more shadow edges to be examined (i.e. A=A∗). By
Lemma 4.1, the total number of iterations is |E(A∗)|−|E(D)| ≤ n2p−m. Further observe that
every operation performed during an iteration requires constant time.
In each iteration, two processes are being carried out.
The first process (Lines 24-27) is the determination of all new shadow corners (if any) of
(t, x) created by (the addition of) the shadow edge ((t, x),(t+ 1, y)) being examined. The total
cost of this process in this iteration is at most two operations for each in-neighbour of (t, y), i.e.,
at most 2c1|Γin
t(y, D)|, where c1∈O(1) is the constant cost of performing a single operation in
this process.
This process is repeated in all iterations, each time with a different shadow edge being
examined. Thus, the cost of 2c1|Γin
t(y, D)|will be incurred for all ((t, x),(t+ 1, y)) ∈E(A∗);
that is, at most ntimes. Summarizing, for each y∈V, t ∈Zpthis process costs 2c1n|Γin
t(y, D)|.
Hence the total cost of this process over all iterations is
X
y∈V,t∈Zp
2c1n|Γin
t(y, D)|= 4 c1n
p−1
X
t=0
mt=O(nm).
The second process, to be performed only if new shadow corners of (t, x) have been found
in the first process, is the determination (Lines 28-33) of all the new shadow edges (if any)
created by the found new shadow corners, and their addition to SE. The cost of this process for
a new shadow corner in this iteration is c2|Γin
t(x, D)|, where c2∈O(1) is the constant cost of
performing a single operation in this process. Observe that, if a new shadow corner of (t, x) is
found in this iteration, it will not be considered in any subsequent iteration (Lines 28-29). Hence,
the cost c2|Γin
t(x, D)|will be incurred at most once for each shadow corner of (t, x); that is, at
most ntimes. Summarizing, for each x∈V, t ∈Zpthis process costs at most 2c2n|Γin
t(x, D)|.
Hence the total cost of this process over all iterations is
X
x∈V,t∈Zp
2c2n|Γin
t(x, D)|= 4 c2n
p−1
X
t=0
mt=O(nm).
18
Consider now the last step of the algorithm, of determining if the constructed Acontains an
anchored star. To determine all the stars (if any) in A∗can be done by checking the degree of
each temporal node in A∗, i.e., in O(np) time. To determine if at least one of them is anchored
can be done by a DFS traversal of A∗starting from each root node (0, x), for a total of at most
O(n2+nm) operations.
It follows that the total cost of the algorithm is O(p n2+nm) as claimed. 2
4.3 Extensions and Improvements
4.3.1 Determining a Copwin Strategy
The algorithm, as described, determines whether or not the arena D(and, thus, the correspond-
ing temporal graph G) is copwin. Simple additions to the algorithm would allow it to easily
determine a copwin strategy σcif Dis copwin.
For any shadow edge e= ((t, x),(t+ 1, y) let ρ(t, x, y) be defined as follows:
1) if e= ((t, x),(t+ 1, y)∈E(D), then ρ(t, x, y) = y.
2) if e= ((t, x),(t+ 1, y)∈E(A∗)\E(D), when eis inserted in S E , either during the
Initialization or the Iteration phase, then ρ(t, x, y) = zwhere (t+ 1, z) is the shadow cover of
(t, y) determined in the corresponding phase of the algorithm (Line 12 if Initialization, Line 28
if Iteration).
Recall that, if Dis copwin, A∗must contain an anchored star, say (t, x), Since (t, x) is a
star, if the cop is located on (t, x) and the robber is located on (t, y), by moving according to
ρ(starting with ρ(t, x, y)) the cop will eventually capture the robber. Since (t, x) is anchored,
it is reachable from some node in G0, say (0, v); that is, there is a journey π((0, v),(t, x)) from
(0, v) to (T , x), where [T]p=t.
Consider now the following strategy σcfor the cop: (1) choose as initial location (0, v);
(2) follow π((0, v),(t, x)); (3) follow ρ. Using this strategy, the cop will eventually capture the
robber.
4.3.2 Improvements
The time costs of the algorithm can be reduced by simple modifications and/or by exploiting
properties of the temporal graphs.
First of all observe that the algorithm can be made to stop as soon as a temporal node
becomes a star (e.g., testing if (t, x) is a star in Line 22) possibly reducing the overall cost with
the early termination.
Observe next that some of the costs of the algorithm can be reduced and some of its processes
simplified if Ghas special properties. Consider for example the properties of reflexivity and
temporal connectivity with respect to the last step of the algorithm, determining if A∗contains
an anchored star.
Property 4.1.
(i) If Gis sourceless then every temporal node of Dis anchored.
(ii) Let Gbe reflexive and temporally connected. If an augmented arena Aof Gcontains a star,
every temporal node of A∗is an anchored star.
19
Proof. (i) Let (t, u) be a temporal node of D; if t= 0, then (t, u) is anchored by definition.
Let t > 0; since, by assumption, no temporal node of Gis a source, there exists a sequence of
edges ej= ((t−j−1, uj−1),(t−j, uj)) ∈E(D) where 0 ≤j≤tand u0=u. In other words
there is a journey from (0, u0) to (t, u), i.e., (t, u) is anchored.
(ii) Let Gbe reflexive and temporally connected and contain a star, say (i, v). Since it is reflexive,
then every snapshot is sourceless, thus, by part (i) of this property, every temporal node of Dis
anchored. Furthermore, it is possible for the cop to wait at any vertex for any amount of time.
This implies that, if the cop reaches vertex vat any time t, it can just wait there until time
[t′]p=i; in other words, all the temporal nodes (j, v), 0 ≤j < p, are star. It also implies that,
if the cop can reach a star from a temporal node (t, u), also that node is star, and so are all its
other temporal instances (j, u), 0 ≤j < p. Since Gis temporally connected, every vertex u∈V
is reachable at some time starting at any time from every other vertex. Thus, the property
holds. 2
The standard game in a static graph assumes the graph to be reflexive and connected. Hence,
if its extension to a periodic graph likewise assumes the graph to be reflexive and temporally
connected, by Property 4.1, the last step of the algorithm would consists of just testing if the
degree of an arbitrary node in A∗is n. That is, instead of O(nm) operations, a single one
suffices.
4.3.3 Game Variations
All our results are established for the unified version of the game. Therefore. all the caracteriza-
tion properties and algorithmic results hold for the standard and for the directed games studied
in the literature, both when the players are restless and when they are not.
They hold also for all those settings (and, thus, variants of the game) not considered in the
literature, where there is a mix of vertices: those where the players must leave and those where
the players can wait; furthermore such a mix might be time-varying (i.e., different in every
round).
5k > 1Cops & Robber
In this section, we consider the C&R game in periodic temporal graphs when there are k > 1
cops. As a first step, we introduce some terminology, and we then show how to extend the
definitions and most of the previous results to this more complex setting.
5.1 Terminology
5.1.1 Configurations
Given a finite multiset (mset) X, we shall denote by ||X|| its wordcount (i.e., the number of its
elements), by X∗its supporting set (i.e., the set of its distinct elements), and by κ(x) the count
(i.e., the multiplicity) of x∈X∗in X.
Given a finite set Pof size |P|=n, a positive integer k, and a mset X, we shall denote
by X⊑kPthe facts that X∗⊆Pand ||X|| =k; we shall say that Xis a mset over Pwith
wordcount k, and denote by [P]kthe collection of all such multisets.
Given an arena D= (Zp×V, E(D)) and a multiset X⊑kV, we shall denote by Γ[t]p(X, D) =
S1≤j≤kΓ[t]p(xj,D) the set of all the outneighbours of the vertices xj∈Xin G[t]p.
20
Aconfiguration is a triple (t, C, r)∈Z+×[V]k×V, where the multiset C=⟨c1, ..., ck⟩ ⊑kV
denotes the positions of the cops and r∈Vthe position of the robber at the beginning of round
t∈Z+. In particular, Cspecifies the number κ(c) of cops in vertex c∈C∗at the beginning of
that round.
The configuration graph CG = (V(CG), E(CG)) of D, defined by
V(CG) = {(t, C, r)|t∈Z+; ([t]p, r)∈V(D); ∀ci∈C, ([t]p, ci)∈V(D)}
E(CG) = {((t, C =⟨c1, ..., ck⟩, r),(t+ 1, C′=⟨c′
1, ..., c′
k⟩, r′))|t∈Z+;c′
i∈Γ[t]p(ci,D) ; r /∈C}
is still acyclic; the source nodes are those with t= 0, the sink nodes are those with r∈C.
A playing strategy for the cops is any function σc:V(CG)→[V]kwhere, for every (t, C, r)∈
V(CG), if r /∈Cthen ((t, C, r),(t+ 1, σc(t, C, r)) ∈E(CG), else σc(t, C, r) = C. It specifies where
the cops should move in round tif they are at ([t]p, C), the robber is at ([t]p, r), and it is the
cops’ turn to move. A playing strategy σrfor the robber is defined in a similar way.
A configuration (t, C, r) is said to be k-copwin if there exists a strategy σcsuch that, starting
from (t, C, r), the cops win the game regardless of the strategy σrof the robber; such a strategy
σcis said to be k-copwin for (t, C, r). A strategy σcis said to be k-copwin if there exists a C
such that σcis k-copwin for (0, C, r) for all r∈V.
If a k-copwin strategy exists, then Gand its arena Dare said to be k-copwin; else they are
k-robberwin.
5.1.2 Directed Multi-Hypergraph
Given a set Vof vertices, a directed multi-hypergraph (dmh) on Vis a pair H= (V, E(H)) where
E(H) is a set of ordered pairs, called hyperedges, of non-empty multisets over V. For hyperedge
e= (V−
e, V +
e)∈ E(H), V−
eand V+
eare called the in-set (or tail) and the out-set (or head) of e,
respectively; ||V−
e|| and ||V+
e|| are called the in-size and the out-size of e, respectively.
A directed multi-hypergraph His homogeneous if all its hyperedges have the same in-size,
and the same out-size, i.e., ∀e, e′∈ E(H),||V−
e|| =||V−
e′|| and ||V+
e|| =||V+
e′||. Since the focus of
our study is on games with kcops and a single robber, in the following we shall consider only
homogeneous directed multi-hypergraphs where, for every hyperedge e∈ E (H), ||V−
e|| =kand
||V+
e|| = 1. For simplicity of notation, since V+
eis a singleton, we shall denote V+
edirectly by
its element. Moreover, we shall denote [V]1simply by V.
5.1.3 Augmented k-Arenas
We can extend the notions of arena and augmented arenas from graphs to hypergraphs in a
direct way.
To each arena D= (V, E (D)) and integer k≥1, there corresponds a unique hypergraph
Dk= (V, E(Dk)), called k-arena defined as follows for all t∈Zpand all x, y ∈V:
((t, x),([t+ 1]p, y)) ∈E(D)⇐⇒ ∀X∈[V]k:x∈X, ((t, X),([t+ 1]p,⟨y⟩)) ∈ E (Dk).
Observe that, like the arena D, the k-arena Dkis composed of pslices St(Dk), t∈Zp; and that
D1is precisely D.
Definition 5.1. An augmented k-arena of D= (V, E(D)) is any hypergraph Ak= (V, E(A))
where:
(1) E(Dk)⊆ E(Ak);
(2) for every t∈Zpand hyperedge e∈ E(Ak), the configuration (t, V −
e, V +
e) is k-copwin in D.
21
We shall refer to the hyperedges of the augmented k-arena Akof Das shadow hyperedges.
Let Ak(D) denote the set of augmented k-arenas of D. Observe that, by definition, Dk∈
Ak(D). Further observe the following.
Property 5.1. The partial order (A(Dk),⊂)induced by hyperedge-set inclusion on A(Dk)is a
complete lattice. Hence (A(Dk),⊂)has a maximum which we denote by Ak
max.
Proof. It follows from the fact that, by definition of augmented k-arena, the set A(Dk) is
closed under the union of augmented k-arenas. 2
Definition 5.2. Let Akbe an augmented k-arena of D. A temporal node (t, y ) is a shadow
k-corner of a multiset X=⟨(t+ 1, x1), ..., (t+ 1, xk)⟩ ⊑kV(D) with ([t+ 1]p, y)/∈X, if
Γt(y, D)⊆Γ[t+1]p(X, Ak).
The multiset Xwill then be called the shadow k-cover of (t, y).
5.2 k-Properties
The equivalent of the three basic properties of augmented arenas established in Section 3, namely
Theorems 3.1, 3.2 and 3.3, can be shown to hold also when k > 1, following analogous proof
arguments.
For simplicity, we shall call a multiset X⊑kV(St) a k-temporal node. A k-temporal node
X=⟨(t, x1),(t, x2), ..., (t, xk)⟩ ⊑kV(St) is said to be a k-star if Γt(X, D) = V. It is said to be
k-anchored if there exists a multiset U=⟨(0, u1),(0, u2), ..., (0, uk)⟩ ⊑kV(D) such that there is
a journey πjfrom (0, uj) to (t, xj) for all 1 ≤j≤k.
The following theorem is a generalization of Theorem 3.1 for kcops.
Theorem 5.1. (k-Characterization Property)
An arena Dis k-copwin if and only if Ak
max contains a k-anchored k-star.
Proof. (if) Let Ak
max contain a k-anchored k-star U=⟨(t, u1),(t, u2), ..., (t, uk)⟩ ⊑kV(D). By
definition of k-star, Γt(U, Dk) = V. Thus, by definition of k-augmented arena, for every v∈V
the configuration (t, U, v) is k-copwin, i.e., there is a k-copwin strategy σcfrom (t, U, v).
Since Uis k-anchored, there exists a k-temporal node X=⟨(0, x1),(0, x2), ..., (0, xk)⟩ ⊑k
V(D) such that there is a journey πjfrom (0, xj) to (t, uj) for all 1 ≤j≤k. Consider now the
cop strategy σ′
cof: (1) initially positioning the kcops c1, c2, ..., ckon the nodes in X, (2) then
each cjmoving according to the journey πj((0, xj),(t, uj)) and, once on (t, uj), (3) following
the copwin strategy σcfrom (t, U, w), where wis the position of the robber at the beginning of
round t. This strategy σ′
cis winning for (0, X, v) for all v∈V; hence Dis k-copwin.
(only if) Let Dbe k-copwin. We show that there must exist an augmented k-arena Akof
Dthat contains a k-anchored k-star. Since Dis copwin, by definition, there must exist some
starting position X=⟨(0, x1),(0, x2), ..., (0, xk)⟩ ⊑kV(D) for the cops such that, for all positions
initially chosen by the robber, the cops eventually capture the robber. In other words, all the
configurations (0, X, v) with v∈Vare k-copwin; thus the k-arena Akobtained by adding to
E(D) the set of hyperedges {(0,⟨x1, x2, ..., xk⟩, v)|v∈V}is an augmented k-arena of Dand the
multiset Xis a k-anchored k-star. By Property 5.1, E(Ak)⊆E(Ak
max) and the theorem follows.
2
The following theorem is a generalization of Theorem 3.2 for kcops.
22
Theorem 5.2. (k-Augmentation Property)
Let Ak∈Ak(D),X=⟨(t, x1), ..., (t, xk)⟩ ⊑kV(D),(t, y)∈V(D)and Z=⟨(t+ 1, z1), ..., (t+
1, zk)⟩ ⊑kV(D). Assume there is a bijection f:X→Zsuch that f(t, xi)is an out-neighbour of
(t, xi)for all 1≤i≤k. If (t, y)is a shadow k-corner of Z, then the arena A′k=Ak∪(X, (t+
1, y)) is an augmented k-arena of D.
Proof. Let Akbe an augmented k-arena of Dand let X=⟨(t, x1), ..., (t, xk)⟩ ⊑kV(D),
(t, y)∈V(D), Z=⟨(t+ 1, z1), ..., (t+ 1, zk)⟩ ⊑kV(D). Assume there is a bijection f:X→Z
such that f(t, xi) is an out-neighbour of (t, xi) for all 1 ≤i≤k, and (t, y) is a shadow k-corner
of Z. The theorem follows if (X, (t+ 1, y)) is already a hyperedge of E(Ak). Consider the case
where (X, (t+ 1, y)) /∈E(Ak). Since (t, y) is a shadow corner of Z, then for every w∈Γt(y, D)
we have that (Z, (t+ 2, w)) ∈E(Ak), i.e., (t+ 1, Z, w) is winning for the cops. If the cops move
from Xto Zusing fwhen the robber is on (t, y), then regardless of the robber’s move, the
resulting configuration would be winning for the cops. In other words, (t, X, y) is a winning
configuration for the cops. It follows that A′k=Ak∪ {(X, (t+ 1, y))}is an augmented arena of
Dk.2
To be able to determine whether the current augmented k-arena of Dis indeed Ak
max, we
can use the following theorem. This is a generalization of Theorem 3.3 for kcops.
Theorem 5.3. (k-Maximality Property)
Let Ak∈Ak(D). Then Ak=Ak
max if and only if, for every hyperedge (X, (t+ 1, y)) /∈E(Ak),
where X⊑kV(St), there exists no Z⊑kV(St+1)for which (1) there exists a bijection f:X→Z
where f(t, xj)is an out-neighbour of (t, xj)for all 1≤j≤k, and (2) (t, y)is a shadow k-corner
of Z.
Proof. (only if ) By contradiction, let Ak=Ak
max but there exists a hyperedge (X, (t+ 1, y)/∈
E(Ak) where X⊑kV(St); and a k-temporal node Z⊑kV(Si) for which (1) there exists bijection
f:X→Zwhere f(t, xj) is an out-neighbour of (t, xj) for all 1 ≤j≤k, and (2) (t, y) is a
shadow k-corner of Z. By Theorem 5.2, A′k=Ak∪(X, (i+ 1, y)) is an augmented arena of D;
however, E(A′k) contains one more edge than E(Ak), contradicting the assumption that Akis
maximum.
(if) Let Ak=Ak
max; that is, there exists (X, (t+ 1, y)) ∈E(Ak
max)\E(Ak). By definition,
the configuration (t, X, y) is k-copwin; let σcbe a copwin strategy for the configuration (t, X, y),
i.e., starting from (t, X, y), the cops win the game regardless of the strategy σrof the robber.
Let C= (V(C), E(C)) ⊆ CG be the directed acyclic graph of configurations induced by σc
starting from (t, X, y), and defined as follows: (1) (t, X, y)∈V(C); (2) if (t′, Y, v)∈V(C)
with t′≥tand v /∈Y, then, for all w∈Γt′(v , D), (t′+ 1, σc(t′+ 1, Y, v), w)∈V(C) and
((t′, Y, v),(t′+ 1, σc(t′+ 1, Y, v), w))) ∈E(C).
Observe that in Cthere is only one source (or root) node, (t, X, y), and every (t′, Z, w)∈V(C)
with w∈Zis a sink (or terminal) node. Since σcis a winning strategy for the root, every node
in Cis a copwin configuration, and every path from the root terminates in a sink node.
Partition V(C) into two sets, Uand Wwhere U={(i, Y, v )|(Y, (i+ 1, v)) ∈E(Ak)}and
W=V(C)\U. Observe that every sink of V(C) belongs to U; on the other hand, since
(X, (t+ 1, y)) /∈E(Ak) by assumption, the root belongs to W.
Given a node κ= (i, Y, v)∈V(C), let C[κ] denote the subgraph of Crooted in κ.
Claim. There exists κ∈V(C)such that all nodes of C[κ], except κ, belong to U.
Proof of Claim. Let P0be the set of sinks of C. Starting from s= 0, consider the set Ps+1
of all in-neighbours of any node of Ps; if Ps+1 does not contain an element of W, then increase
23
General Strategy (k≥1)
1. While there is a still unexamined shadow hyperedge e= (X, (t+ 1, y)) ∈E(Ak) do:
2. If there are still unexamined shadow k-corners covered by Xthen:
3. For each such shadow k-corner (t−1, z) do:
4. If there are new shadow hyperedges due to (t−1, z) then:
5. Add them to Akto be examined.
6. Remove (t−1, z) from consideration as a shadow corners of X(i.e., mark it as examined).
7. Remove efrom consideration (i.e., mark it as examined).
8. If there is an star, then Dis copwin else it is robberwin.
Figure 7: Outline of general strategy; it terminates when Ak=Ak
max.
sand repeat the process. Since (t, X, y )∈W, this process terminates for some s≥0, and the
Claim holds for every κ∈Ps+1 ∩W.2
Let (t′, X′, y′) be a node of V(C) satisfying the above Claim. Thus (X′,(t′+ 1, y′)) /∈E(Ak)
but, since (i′, X ′, y′) is copwin, (X′,(t′+ 1, y′)) ∈ Ak
max. By the Claim, all other nodes of
C[(t′, X′, y′)] belong to U, in particular the set of nodes {(t′+ 1, B, z)|B=σc(t′, X ′, y′), z ∈
Γt′(y′,D)}. This means that, for every z∈Γt′(y′,D), (t′+ 1, B, z)∈E(Ak). In other words,
Γt′(y′,D)⊆Γt′+1 B, Ak; that is, (t′, y′) is a shadow corner of B.
Summarizing: by assumption Ak=Ak
max; as shown, (X′,(t′+ 1, y′)) ∈E(Ak
max)\E(Ak),
and B⊆Γt′(X′,D) is a shadow cover of (t′, y′); that is, Γt′(y′,D)⊆Γt′+1 B , Ak, concluding
the proof of the if part of the theorem. 2
5.3 k-Copwin Determination
Based on the characterization and properties of k-copwin periodic graphs established in the pre-
vious section, a general strategy to determine if a peridic graph is k-copwin follows directly (see
Figure 7) along the same lines of the one discussed in Section 4.2. The immediate implementa-
tion of the strategy does lead to a solution algorithm with proof of correctness and complexity
analysis following exactly the same lines of that for k= 1; its time complexity however does not
improve the O(k p nk+2) bound reported in [41].
6 Conclusions
In this paper we have provided a complete characterization of copwin periodic temporal graphs,
establishing several basic properties on the nature of a copwin game in such graphs.
These characterization results are general, in the sense that they do not rely on any assump-
tion on properties such as connectivity, symmetry, reflexivity held (or not held) by the individual
static graphs in the periodic sequence defining the temporal graph.
These results have been established for the unified version of the game, which includes the
standard undirected and directed versions of the game, both in the original and restless variants,
as well as new variants never studied before.
Based on these results, we have also designed an algorithm that determines if a temporal
graph with period pis copwin in time O(p n2+nm), where mis the number of edges in the
24
arena, improving the (then) existing bounds for periodic and static graphs. Observe that it
has been recently shown in [41] that our bound can be reached also by using the reduction to
reachability games of [20].
In the case k > 1 of multiple cops, by shifting from a representation in terms of directed
graphs to one in terms of directed multi-hypergraphs, we proved that all the fundamental prop-
erties of augmented arenas established for k= 1 continue to hold, providing a complete charac-
terization of k-copwin periodic graphs.
The established results for k > 1 lead directly to a solution strategy to determine if a
periodic temporal graph is k-copwin; however, the immediate straightforward implementation
of the strategy appears to achieve a less efficient results that the reported O(k p nk+2) bound
reported in [41]. An outstanding open problem is whether, using the characterization properties
established here, it is possible to match if not improve the existing bound.
Other than the algorithmic aspects, a major open research direction is the study of structural
properties of copwin temporal graphs, extending to the temporal realm the extensive investiga-
tions carried out on static graphs. This investigation has just started [7].
Acknowledgments
This research has been supported in part by the Natural Sciencies and Engineering Research
Council of Canada (NSERC) under the Discovery Grant program. Some of these results have
been presented at the 30th International Colloquium on Structural Information and Communi-
cation Complexity [8].
References
[1] A. Agarwalla, J. Augustine, W. Moses, S. Madhav, and A. K. Sridhar. Deterministic dis-
persion of mobile robots in dynamic rings. In 19th International Conference on Distributed
Computing and Networking, pages 19:1–19:4, 2018.
[2] M. Aigner and M. Fromme. A game of cops and robbers. Discrete Applied Mathematics,
(8):1–12, 1984.
[3] S. Balev, J. L. J. Laredo, I. Lamprou, Y. Pign´e, and E. Sanlaville. Cops and robbers on
dynamic graphs: Offline and online case. In 27th International Colloquium on Structural
Information and Communication Complexity (SIROCCO), 2020.
[4] A. Bonato and G. MacGillivray. Characterizations and algorithms for generalized Cops and
Robbers games. Contributions to Discrete Mathematics, 12(1), 2017.
[5] A. Bonato and R. Nowakowski. The Game of Cops and Robbers on Graphs. American
Mathematical Society, 2011.
[6] M. Bournat, S. Dubois, and F. Petit. Computability of perpetual exploration in highly
dynamic rings. In 37th IEEE International Conference on Distributed Computing Systems,
pages 794–804, 2017.
[7] J.-L. De Carufel, P. Flocchini, N. Santoro, and F. Simard. Copnumbers of periodic graphs.
In 54th Southeastern Conference on Combinatorics, Graph Theory, and Computing (in
press). Springer, 2023.
25
[8] J.-L. De Carufel, P. Flocchini, N. Santoro, and F. Simard. Cops & robber on periodic
temporal graphs: Characterization and improved bounds. In 30th International Colloquium
on Structural Information and Communication Complexity (SIROCCO), pages 386–405,
2023.
[9] A. Casteigts, P. Flocchini, B. Mans, and N. Santoro. Measuring temporal lags in delay-
tolerant networks. IEEE Transactions on Computer, 63(2):397–410, 2014.
[10] A. Casteigts, P. Flocchini, W. Quattrociocchi, and N. Santoro. Time-varying graphs and
dynamic networks. Int. J. Parallel Emergent Distributed Syst., 27(5):387–408, 2012.
[11] A. Casteigts, R. Klasing, Y.M. Neggaz, and J.G. Peters. Efficiently testing T-interval
connectivityy in dynamic graphs. In 9th International Conference on Algorithms and Com-
plexity, pages 89–100, 2015.
[12] A. Casteigts, R. Klasing, Y.M. Neggaz, and J.G. Peters. Computing parameters of sequence-
based dynamic graphs. Theory Comput. Syst, 63(3):394–417, 2019.
[13] S. Das and H. Gahlawat. On the cop number of string graphs. In 33rd International
Symposium on Algorithms and Computation (ISAAC), pages 45:1–45:18, 2022.
[14] S. Das, H. Gahlawat, U.k. Sahoo, and S. Sen. Cops and robber on some families of oriented
graphs. Theoretical Computer Science,, 888:31–40, 2021.
[15] G. A. Di Luna. Mobile Agents on Dynamic Graphs, Chapter 20 of [23]. Springer, 2019.
[16] G. A. Di Luna, S. Dobrev, P. Flocchini, and N. Santoro. Distributed exploration of dynamic
rings. Distributed Computing, 33:41–67, 2020.
[17] G. A. Di Luna, P. Flocchini, L. Pagli, G. Prencipe, N. Santoro, and G. Viglietta. Gathering
in dynamic rings. Theoretical Computer Science, 811:79–98, 2020.
[18] T. Erlebach, M. Hoffmann, and F. Kammer. On temporal graph exploration. In 42nd
International Colloquium on Automata, Languages, and Programming, pages 444–455, 2015.
[19] T. Erlebach and J. T. Spooner. Faster exploration of degree-bounded temporal graphs. In
43rd International Symposium on Mathematical Foundations of Computer Science (MFCS),
pages 1–13, 2018.
[20] T. Erlebach and J. T. Spooner. A game of cops and robbers on graphs with periodic edge-
connectivity. In 46th International Conference on Current Trends in Theory and Practice
of Informatics (SOFSEM), pages 64–75, 2020.
[21] A. Ferreira. Building a reference combinatorial model for MANETs. IEEE Network,
18(5):24–29, 2004.
[22] P. Flocchini, B. Mans, and N. Santoro. On the exploration of time-varying networks.
Theoretical Computer Science, 469:53–68, 2013.
[23] P. Flocchini, G. Prencipe, and N. Santoro (Eds.).Distributed Computing by Mobile Entities.
Springer, 2019.
26
[24] T. Gotoh, P. Flocchini, T. Masuzawa, and N. Santoro. Exploration of dynamic networks:
Tight bounds on the number of agents. Journal of Computer and System Science, 122:1–18,
2021.
[25] T. Gotoh, Y. Sudo, F. Ooshita, H. Kakugawa, and T. Masuzawa. Group exploration of
dynamic tori. In IEEE 38th International Conference on Distributed Computing Systems
(ICDCS), pages 775–785, 2018.
[26] I. Gromovikov, W. Kinnersley, and B. Seamone. Fully active cops and robbers. Australasian
Journal of Combinatorics, 76(2):248–265, 2020.
[27] F. Harary and G. Gupta. Dynamic graph models. Mathematical and Computer Modelling,
25(7):79–88, 1997.
[28] P. Holme and J. Saram¨aki. Temporal networks. Physics Reports, 519(3):97–125, 2012.
[29] D. Ilcinkas, R. Klasing, and A. M. Wade. Exploration of constantly connected dynamic
graphs based on cactuses. In 21st International Colloquium Structural Information and
Communication Complexity (SIROCCO), pages 250–262, 2014.
[30] D. Ilcinkas and A. M. Wade. On the power of waiting when exploring public transportation
systems. In 15th International Conference on Principles of Distributed Systems (OPODIS),
pages 451–464, 2011.
[31] R. Jathar, V. Yadav, and A. Gupta. Using periodic contacts for efficient routing in delay
tolerant networks. Ad Hoc & Sensor Wireless Networks, 22(1,2):283–308, 2014.
[32] D. Khatri, N. Komarov, A. Krim-Yee, N. Kumar, B. Seamone, V. Virgile, and A. Xu. A
study of cops and robbers in oriented graphs. ArXiv:1811.06155, 2018.
[33] F. Kuhn, N. Lynch, and R. Oshman. Distributed computation in dynamic networks. In
42nd ACM Symposium on Theory of Computing, pages 513–522, 2010.
[34] P.-S. Loh and S. Oh. Cops and robbers on planar-directed graphs. Journal of Graph Theory,
86(3):329–340, 2017.
[35] N. Morawietz, C. Rehs, and M. Weller. A timecop’s work is harder than you think. In
45th International Symposium on Mathematical Foundations of Computer Science (MFCS),
pages 71:1–14, 2020.
[36] N. Morawietz and P. Wolf. A timecop’s chase around the table. In 46th International
Symposium on Mathematical Foundations of Computer Science (MFCS), pages 77:1–18,
2021.
[37] R. Nowakowski and P. Winkler. Vertex-to-vertex pursuit in a graph. Discrete Mathematics,
43(2-3):235–239, 1983.
[38] J. Petr, J. Portier, and L. Versteegen. A faster algorithm for cops and robbers. Discrete
Applied Mathematics, 320:11–14, 2022.
[39] A. Quilliot. Jeux et points fixes sur les graphes. Ph.D. Thesis. Universit´e de Paris VI, 1978.
[40] F. Simard. Temporal distances and cops and robber games on temporal networks. PhD
thesis, University of Ottawa, 2024.
27
[41] J. T. Spooner T. Erlebach, N. Morawietz and P. Wolf. A cop and robber game on edge-
periodic temporal graphs. J. Comput. Syst. Sci., 144:103534, 2024.
[42] K. Wehmuth, A. Ziviani, and E. Fleury. A unifying model for representing time-varying
graphs. In 2nd IEEE International Conference on Data Science and Advanced Analytics,
(DSAA), pages 1–10, 2015.
28
