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Cops and Robbers on -Free Graphs
... The Meyniel's conjecture is a famous open question that states that for any graph G, the cop number c(G) = O( √ n) [BB13]. The problem has been studied from different perspectives: on various classes of graphs [CNST24,JKT10], characterizations [CM12], and variants of the original problem [CCNV11,JSU23]. ...
In the cops and robber game, there are multiple cops and a single robber taking turns moving along the edges of a graph. The goal of the cops is to capture the robber (move to the same vertex as the robber) and the goal of the robber is to avoid capture. The cop number of a given graph is the smallest number of cops required to ensure the capture of the robber. The k-component order connectivity of a graph G = (V, E) is the size of a smallest set U, such that all the connected components of the induced graph on V \ U are of size at most k. In this brief note, we provide a bound on the cop number of graphs in terms of their 2-component order connectivity.
Andreae proved that the cop number of connected ‐minor‐free graphs is bounded for every graph . In particular, the cop number is at most if contains no isolated vertex, where . The main result of this paper is an improvement on this bound, which is most significant when is small or sparse, for instance, when can be obtained from another graph by multiple edge subdivisions. Some consequences of this result are improvements on the upper bound for the cop number of ‐minor‐free graphs, ‐minor‐free graphs and linklessly embeddable graphs.
In a game of Cops and Robbers on graphs, usually the cops' objective is to capture the robber---a situation which the robber wants to avoid invariably. In this paper, we begin with introducing the notions of trapping and confining the robber and discussing their relations with capturing the robber. Our goal is to study the confinement of the robber on graphs that are free of a fixed path as an induced subgraph. We present some necessary conditions for graphs G not containing the path on k vertices (referred to as -free graphs) for some , so that cops do not have a strategy to capture or confine the robber on G (Propositions 2.1, 2.3). We then show that for planar cographs and planar -free graphs the confining cop number is at most one and two, respectively (Corollary 2.4). We also show that the number of vertices of a connected cograph on which one cop does not have a strategy to confine the robber has a tight lower bound of eight. Moreover, we explore the effects of twin operations---which are well known to provide a characterization of cographs---on the number of cops required to capture or confine the robber on cographs. Finally, we pose two conjectures on confining the robber on -free graphs and the smallest planar graph of confining cop number of three.
Graphs.- Groups.- Transitive Graphs.- Arc-Transitive Graphs.- Generalized Polygons and Moore Graphs.- Homomorphisms.- Kneser Graphs.- Matrix Theory.- Interlacing.- Strongly Regular Graphs.- Two-Graphs.- Line Graphs and Eigenvalues.- The Laplacian of a Graph.- Cuts and Flows.- The Rank Polynomial.- Knots.- Knots and Eulerian Cycles.- Glossary of Symbols.- Index.
The problem of computing the chromatic number of a P
5-free graph (agraph which contains no path on 5 vertices as an induced subgraph) is known to be NP-hard. However, we show
that for every fixed integer k, there exists a polynomial-time algorithm determining whether or not a P
5-free graph admits a k-coloring, and finding one, if it does.
The game of cops and robber is a turn-based vertex pursuit game played on a connected graph between a team of cops and a single robber. The cops and the robber move alter-nately along the edges of the graph. We say that the team of cops wins the game if a cop and the robber are at the same vertex of the graph. The minimum number of cops required to win in a connected graph is called the cop number of the graph. Sivaraman (2019) [16]conjectured that for every t≥5, the cop number of a connected Pt-free graph is at most t−3, where Ptdenotes a path on tvertices. Turcotte (2022) [18]showed that the cop number of any 2K2-free graph is at most 2, which was earlier conjectured by Sivaraman and Testa. Note that if a connected graph is 2K2-free, then it is also P5-free. Liu showed that the cop number of a connected (Pt, H)-free graph is at most t−3, where His a cy-cle of length at most tor a claw. So the conjecture of Sivaraman is true for (P5, H)-free graphs, where His a cycle of length at most 5or a claw. In this paper, we show that the cop number of a connected (P5, H)-free graph is at most 2, where H∈{diamond, paw, K4, 2K1∪K2, K3∪K1}.
We study relationships between the diameter D(G), the domination number γ(G), the independence number α(G) and the cop number c(G) of a connected graph G, showing (i.) c(G)≤α(G)−⌊D(G)−32⌋, and (ii.) c(G)≤γ(G)−D(G)3+O(D(G)).
We prove that the cop number of any 2K2-free graph is at most 2, proving a conjecture of Sivaraman and Testa. We also show that the upper bound of 3 on the cop number of 2K1+K2-free (co-diamond–free) graphs is best possible.
It is known that the class of all graphs not containing a graph H as an induced subgraph is cop-bounded if and only if H is a forest whose every component is a path [4]. In this paper, we characterize all sets H of graphs with bounded diameter, such that H-free graphs are cop-bounded. This, in particular, gives a characterization of cop-bounded classes of graphs defined by a finite set of connected graphs as forbidden induced subgraphs. Furthermore, we extend our characterization to the case of cop-bounded classes of graphs defined by a set H of forbidden graphs such that the components of members of H have bounded diameter.
We adapt the Gyárfás path argument to prove that t−2 cops can capture a robber, in at most t−1 moves, in the game of Cops and Robbers played in a graph that does not contain the t-vertex path as an induced subgraph.
The cop number c(G) of a graph G is an invariant connected with the genus and the girth. We prove that for a fixed k there is a polynomial-time algorithm which decides whether c(G) ≤ k. This settles a question of T. Andreae. Moreover, we show that every graph is topologically equivalent to a graph with c ≤ 2. Finally we consider a pursuit-evasion problem in Littlewood′s miscellany. We prove that two lions are not always sufficient to catch a man on a plane graph, provided the lions and the man have equal maximum speed. We deal both with a discrete motion (from vertex to vertex) and with a continuous motion. The discrete case is solved by showing that there are plane graphs of cop number 3 such that all the edges can be represented by straight segments of the same length.
This note treats the existence of connected, undirected graphs homogeneous of degree d and of diameter k, having a number of nodes which is maximal according to a certain definition. For k = 2 unique graphs exist for d = 2, 3, 7 and possibly for d = 57 (which is undecided), but for no other degree. For k = 3 a graph exists only for d = 2. The proof exploits the characteristic roots and vectors of the adjacency matrix (and its principal submatrices) of the graph.
In a previous paper, Aigner and Fromme (Discrete Appl. Math.8 (1984), 1–12) considered a game played on a finite connected graph G where s pursuers try to catch one evader. They introduced c(G) as the minimal number s of pursuers that are sufficient to catch the evader and showed that, in general, c(G) can be arbitrarily high. On the other hand, they proved that c(G) ≤ 3 if G is planar. The present paper relates c(G) to the “forbidden minor concept.” Suppose that the graph H is not a minor of G and that, for a vertex hϵV(H) H − h has no isolated vertices. It is shown that this implies c(G) ≤ |E(H − h)|. As a consequence, one finds that, for each graph H, there exists a minimal positive integer α(H) such that c(G) ≤ α(H) when H is not a minor of G and, in addition, α(H) < |E(H)| for each connected H with at least two edges. These results are refined by proving that α(K5) = α(K3,3) = 3 and α(K5−) = α(K3,3−) = 2, thereby also refining the above result on planar graphs. (K5−(K3,3−) denotes K5(K3,3) minus an edge.) Further, , where Wn is the wheel with n rim vertices (n ≥ 3). In addition, we also establish an upper bound for c(G) in terms of the cross-cap number of G, thus providing a (partial) analogue to a result of Quilliot on the genus. Finally, the relationship between c(G) and simplicial decompositions is studied and a list of open problems is presented.
A graph G is given and two players, a cop and a robber, play the following game: the cop chooses a vertex, then the robber chooses a vertex, then the players move alternately beginning with the cop. A move consists of staying at one's present vertex or moving to an adjacent vertex; each move is seen by both players. The cop wins if he manages to occupy the same vertex as the robber, and the robber wins if he avoids this forever.We characterize the graphs on which the cop has a winning strategy, and connect the problem with the structure theory of graphs based on products and retracts.
Let G be a finite connected graph. Two players, called cop C and robber R, play a game on G according to the following rules. First C then R occupy some vertex of G. After that they move alternately along edges of G. The cop C wins if he succeeds in putting himself on top of the robber R, otherwise R wins. We review an algorithmic characterization and structural description due to Nowakowski and Winkler. Then we consider the general situation where n cops chase the robber. It is shown that there are graphs on which arbitrarily many cops are needed to catch the robber. In contrast to this result, we prove that for planar graphs 3 cops always suffice to win.
In this paper we will consider Ramsey-type problems for finite graphs, r-partitions and hypergraphs. All these problems ask for the existence of large homogeneous (monochromatic) configurations of a certain kind under the condition that the size of the underlying set is large. As it is quite common in Ramsey theory, most of our results are not sharp and almost all of them lead to new problems which seem to be difficult. The problems we treat are only loosely connected. So we will state and explain them section by section.