Three ways to cover a graph
ABSTRACT We consider the problem of covering a host graph $G$ with several graphs from
a fixed template class ${T}$. The classical covering number of $G$ with respect
to ${T}$ is the minimum number of template graphs needed to cover the edges of
$G$. We introduce two new parameters: the local and the folded covering number.
Each parameter measures how far $G$ is from the template class in a different
way. Whereas the folded covering number has been investigated thoroughly for
some template classes, e.g., interval graphs and planar graphs, the local
covering number was given only little attention.
We provide new bounds on each covering number w.r.t. the following template
classes: linear forests, star forests, caterpillar forests, and interval
graphs. The classical graph parameters turning up this way are intervalnumber,
tracknumber, and linear, star, and caterpillar arboricity. As host graphs we
consider graphs of bounded degeneracy, bounded degree, or bounded (simple)
treewidth, as well as, outerplanar, planar bipartite and planar graphs. For
several pairs of a host class and a template class we determine the maximum
(local, folded) covering number of a host graph w.r.t. that template class
exactly.
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Article: Multilayer Grid Embeddings for VLSI.
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ABSTRACT: In this paper we propose two new multilayer grid models for VLSI layout, both of which take into account the number of contact cuts used. For the first model in which nodes “exist” only on one layer, we prove a tight area × (number of contact cuts) = Θ(n 2) tradeoff for embeddingnnode planar graphs of bounded degree in two layers. For the second model in which nodes “exist” simultaneously on all layers, we give a number of upper bounds on the area needed to embed groups using no contact cuts. We show that anynnode graph of thickness 2 can be embedded on two layers inO(n 2) area. This bound is tight even if more layers and any number of contact cuts are allowed. We also show that planar graphs of bounded degree can be embedded on two layers inO(n 3/2(logn)2) area. Some of our embedding algorithms have the additional property that they can respect prespecified grid placements of the nodes of the graph to be embedded. We give an algorithm for embeddingnnode graphs of thicknessk ink layers usingO(n 3) area, using no contact cuts, and respecting prespecified node placements. This area is asymptotically optimal for placementrespecting algorithms, even if more layers are allowed, as long as a fixed fraction of the edges do not use contact cuts. Our results use a new result on embedding graphs in a singlelayer grid, namely an embedding ofnnode planar graphs such that each edge makes at most four turns, and all nodes are embedded on the same line.Algorithmica 01/1991; 6:129151. · 0.49 Impact Factor  Mathematica Slovaca 01/1980; · 0.39 Impact Factor
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Article: Scheduling Split Intervals.
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ABSTRACT: We consider the problem of scheduling jobs that are given as groups of nonintersecting segments on the real line. Each job J j is associated with an interval, I j , which consists of up to t segments, for some t 1, a positive weight, w j , and two jobs are in conflict if any of their segments intersect. Such jobs show up in a wide range of applications, including the transmission of continuousmedia data, allocation of linear resources (e.g. bandwidth in linear processor arrays), and in computational biology/geometry. The objective is to schedule a subset of nonconflicting jobs of maximum total weight.SIAM J. Comput. 01/2006; 36:115.
Page 1
arXiv:1205.1627v1 [math.CO] 8 May 2012
Three ways to cover a graph
Kolja Knauer1⋆and Torsten Ueckerdt2⋆⋆
1Technical University Berlin, Berlin, Germany
knauer@math.tuberlin.de
2Charles University in Prague, Prague, Czech Republic
ueckerdt@googlemail.com
Abstract. We consider the problem of covering a host graph G with
several graphs from a fixed template class T . The classical covering num
ber of G with respect to T is the minimum number of template graphs
needed to cover the edges of G. We introduce two new parameters: the
local and the folded covering number. Each parameter measures how far
G is from the template class in a different way. Whereas the folded cover
ing number has been investigated thoroughly for some template classes,
e.g., interval graphs and planar graphs, the local covering number was
given only little attention.
We provide new bounds on each covering number w.r.t. the following
template classes: linear forests, star forests, caterpillar forests, and in
terval graphs. The classical graph parameters turning up this way are
intervalnumber, tracknumber, and linear, star, and caterpillar arboric
ity. As host graphs we consider graphs of bounded degeneracy, bounded
degree, or bounded (simple) treewidth, as well as, outerplanar, planar
bipartite and planar graphs. For several pairs of a host class and a tem
plate class we determine the maximum (local, folded) covering number
of a host graph w.r.t. that template class exactly.
1Introduction
Graph covering is one of the most classical topics in graph theory. In 1891, in
one of the first purely graphtheoretical papers at all, Petersen shows that any
2rregular graph can be covered with r sets of vertex disjoint cycles [41]. A first
survey on covering problems of Beineke [10] appeared in 1969. Graph covering
problems are lively and ramified fields of research – over the last decades as well
as today [28,29,2,3,24,39]. This is supported through the course of this paper by
many references to recent works of different authors.
In every graph covering problem one is given a host graph G, a template class T ,
and a notion of how to cover G with one or several template graphs. One is then
interested in covers of G w.r.t. T that are in some sense simple, or well structured;
the most prevalent measure of simplicity being the number of template graphs
needed to cover G.
⋆Partially supported by DFG grant FE340/81 as part of ESF project GraDR EU
ROGIGA
⋆⋆Research was supported by GraDR EUROGIGA project No. GIG/11/E023
Page 2
The main motivation of this paper is to introduce the following three parameters,
each of which represents how well G can be covered w.r.t. T in a different way:
The global covering number, or simply covering number, is the most classical one.
All kinds of arboricities, e.g. star [4], caterpillar [23], linear [3], pseudo [42], and
ordinary [40] arboricity of a graph are global covering numbers. Other global
covering numbers are the (outer) thickness [10,39] and the tracknumber [26]
of a graph. To the best of our knowledge the only local covering number in
the literature is the bipartite degree introduced by Fishburn and Hammer [20].
Here the coloringaspect is removed from the global covering number, but the
underlying covering problem is the same. Finally, the folded covering number
underlies a different, but related, concept of covering. It has been investigated
w.r.t. interval graphs and planar graphs as template class. In the former case
the folded covering number is known as the intervalnumber [30], in the latter
case as the splittingnumber [32] of a graph.
While some covering numbers, like arboricities, are of mainly theoretical interest,
others, like thickness, intervalnumber and tracknumber have wide applications
in VLSI design [1], network design [43], scheduling and resource allocation [9,12],
and bioinformatics [35,33]. The three covering numbers presented here not only
unify the notion in the literature, they as well seem interesting in their own right,
e.g., provide new approaches to attack or support classical open problems.
In this paper we moreover present new lower and upper bounds for several cov
ering numbers, in particular w.r.t. the template classes: interval graphs, star
forests, linear forests, and caterpillar forests, and host classes: graphs of bounded
degeneracy or bounded (simple) treewidth, as well as outerplanar, planar bi
partite, planar, and regular graphs. We provide an overview over some of our
new results in Table 1.
star forests
g
caterpillar forests
g
ℓ
interval graphs
ℓℓ = f
fg
f
outer
planar
planar +
bipartite
planar
stw ≤ k
tw ≤ k
deg ≤ k
3 [27]3
3 [36]
3
3
2 [36]
2
2 [45]
4 (T.9)
3 (C.2)
4 [22]
3
34
3
3 [45]
5 [5,27]
4 (C.2)
4 [22]
4
4 [45]
4 [23]
?3 [45]
k (T.6)
k + 1
k + 1
k + 1 k + 1 k + 1 (T.8) k + 1 (T.7) k (T.5)
k + 1 [15]
2k [6,27] k + 1 (C.1)
k + 1
k + 1 k + 1
k + 1
k + 1
2k (T.4)
k + 1 k + 1 (T.6)
2k
k + 1
k + 1
k + 1
k + 1
Table 1. Overview of results. Each row of the table corresponds to a host class G, each
column to a template class T . Every cell contains the maximum covering number of
all graphs G ∈ G w.r.t. the template class T , where the columns labeled g,ℓ,f stand
for the global, local and folded covering number, respectively. Grey entries follow by
Proposition 1 from other stronger results in the table. Letters T. and C. stand for
Theorem and Corollary in the present paper, respectively.
Page 3
This paper is structured as follows: In order to give a motivating example before
the general definition, we start by discussing in Section 2 the linear arboricity
and its local and folded variants. In Section 3 the three covering numbers are
formally introduced and some general properties are established. In Section 4
we introduce the template classes star forests, caterpillar forests, and interval
graphs, and in Section 5 we present our results claimed in Table 1. In Section 6 we
briefly discuss the computational complexity of some covering numbers, giving
a polynomial time algorithm for the local star arboricity. Moreover, we discuss
by how much global, local and folded covering numbers can differ.
2Folded and Local Linear Arboricity
We give the general definitions of covers and covering numbers in Section 3
below. In this section we motivate and illustrate these concepts on the basis of
one fixed template class: the class L of linear forests, i.e., every graph L ∈ L is
the disjoint union of simple paths. We want to cover a host graph G = (V,E)
by several linear forests L1,...,Lk∈ L, i.e., every edge e ∈ E shall be contained
in at least3one Liand no nonedge of G shall be contained in any Li. If G is
covered by L1,...,Lkwe denote this by G =?
The linear arboricity of G, denoted by la(G), is the minimum k such that G =
?
degree ∆ has la(G) ≥ ⌈∆
In 1980, Akiyama et. al. [3] stated the Linear Arboricity Conjecture (LAC). It
says that the linear arboricity of any simple graph of maximum degree ∆ is
either ⌈∆
2⌉. LAC is confirmed for planar graphs by Wu and Wu [50,51]
and asymptotically for general graphs by Alon and Spencer [7]. The general
conjecture remains open. The bestknown upper bound for la(G) is ⌈3∆+2
to Guldan [25].
We define the local linear arboricity of G, denoted by laℓ(G), as the minimum
j such that G =?
different Li. Again if G has maximum degree ∆ then laℓ(G) ≥ ⌈∆
is ∆regular then laℓ(G) ≥ ⌈∆+1
the following statement must necessarily hold for LAC to be true.
i∈[k]Li.
i∈[k]Liand Li∈ L for i ∈ [k]. One easily sees that every graph G of maximum
2⌉, and every ∆regular graph G has la(G) ≥ ⌈∆+1
2⌉.
2⌉ or ⌈∆+1
5
⌉, due
i∈[k]Li and every vertex v in G is contained in at most j
2⌉, and if G
2⌉. Note that laℓ(G) is at most la(G) and hence
Conjecture 1. Local Linear Arboricity Conjecture (LLAC): The local linear ar
boricity of any simple graph with maximum degree ∆ is either ⌈∆
2⌉ or ⌈∆+1
2⌉.
Observation 1 To prove LAC or LLAC it would suffice to consider ∆regular
graphs with ∆ odd: Regularity is obtained by considering a ∆regular supergraph
of G. If ∆ is even, say ∆ = 2k, one can find a spanning linear forest Lk+1 in
G [25], remove it from the graph, and complement Lk+1 by a cover L1,...,Lk
in the remaining graph on maximum degree ∆ − 1 = 2k − 1.
3Since linear forests are closed under taking subgraphs, we can indeed assume that
e ∈ Li for exactly one i ∈ [k].
Page 4
If G is ∆regular with ∆ odd, then LLAC states that G =?
vertex being an endpoint of exactly one path. While LAC additionally requires
that the paths can be colored with ⌈∆
2⌉ colors such that no two paths that share
a vertex receive the same color. We will see in later sections that sometimes the
coloring is the crucial and difficult task.
Next we propose a second way to cover the host graph G with linear forests. A
walk in G is a not necessarily edgedisjoint path. As before, a set W1,...,Wkof
walks covers G, denoted by G =?
of the edgesets of the walks. We are now interested in how often a vertex v
in G appears in the walks W1,...,Wk in total. The folded linear arboricity of
G, denoted by laf(G), is the minimum j such that G =?
vertex v in G appears in total at most j times in the walks W1,...,Wk. Again
if G has maximum degree ∆ then laf(G) ≥ ⌈∆
laf(G) ≥ ⌈∆+1
from a theorem of West [48] (where it is stated in terms of the intervalnumber
i(G)). It is a weakening of LLAC above.
i∈[k]Liwith every
i∈[k]Wi, if the edgeset E of G is the union
i∈[k]Wi and every
2⌉, and if G is ∆regular then
2⌉. Clearly, laf(G) ≤ laℓ(G). The next theorem follows directly
Theorem 2. If G has maximum degree ∆ then laf(G) ∈ {⌈∆
2⌉,⌈∆+1
2⌉}.
Proof. If G is not Eulerian add a vertex x to G and connect it to every vertex in
G of odd degree. Consider any Eulerian tour in G ∪ x (or G) and (if necessary)
split it into shorter walks by removing x from it.
⊓ ⊔
Remark 1. Besides LLAC, there are several more weakenings of LAC that are
still open. For example4, it is open whether the caterpillar arboricity of maximum
degree ∆ graphs is at most ⌈∆+1
2⌉. Yet a weaker, but still open, question asks
whether the tracknumber of these graphs is at most ⌈∆+1
2⌉.
3 Covers and Covering Numbers
We formalize the concepts from Section 2 w.r.t. general template classes. For
a host graph G and a template class T , we define a cover of G w.r.t. T to be
an edgesurjective homomorphism ϕ from the disjoint union T1· ∪T2· ∪··· · ∪Tkof
template graphs, i.e., Ti∈ T for i ∈ [k], to G. The size of a cover is the number
of template graphs in the disjoint union. A cover ϕ is called injective if ϕTi,
that is, ϕ restricted to Ti, is injective for every i ∈ [k].
Definition 1. For a template class T and a host graph G = (V,E) define the
(global) covering number cT
covering number cT
f(G) as follows:
g(G), the local covering number cT
ℓ(G), and the folded
cT
cT
g(G) = min{size of ϕ : ϕ injective cover of G w.r.t. T }
ℓ(G) = min?maxv∈Vϕ−1(v) : ϕ injective cover of G w.r.t. T?
4See Section 4.2 and 4.3 for the definition of caterpillar arboricity and tracknumber,
respectively.
Page 5
cT
f(G) = min?maxv∈Vϕ−1(v) : ϕ cover of G w.r.t. T of size 1?
Let us rephrase cT
number of template graphs needed to cover the host graph, where covering means
identifying subgraphs in G that are template graphs, such that every edge of G
is contained in some template graph. In the local covering number the number of
template graphs in such a cover is not restricted; Instead the number of template
graphs at every vertex should be small. We will see later (and already indicated
in Section 2) that these two numbers can differ significantly. The folded covering
number is the minimum k such that every vertex v of G can be split into at most k
vertices, distributing the incident edges at v arbitrarily (even repeatedly) among
them, such that the resulting graph belongs to the template class.
Within the scope of this paper we only consider template classes that are closed
under disjoint union even without explicitly saying so. E.g., when considering,
say stars or cliques, as template graphs we actually mean star forests and col
lections of cliques, respectively. If the template class T is closed under disjoint
union, then the restriction to covers of size 1 in the definition of cT
essary. This property is also needed to prove the inequality cT
Proposition 1 (i).
g(G), cT
ℓ(G), and cT
f(G): The covering number is the minimum
fis unnec
f(G) in
ℓ(G) ≥ cT
Remark 2. It is still interesting to consider template classes that are not closed
under disjoint union. For example, if Cy′denotes the class of all simple cycles,
then Haj´ os’ Conjecture [37] states that for any nvertex Eulerian graph G we
have cCy′
g (G) ≤ ⌊n
2⌋. (It is easily seen that cCy′
ℓ(G), cCy′
f(G) ≤ ⌊n
2⌋.)
For any graph parameter one is usually interested in its maximum (or minimum)
value within certain graph classes. For i = g,ℓ,f, a template class T and a graph
class G, called the host class, we define cT
this section with a list of inequalities, all of which are elementary applications
of Definition 1 and homomorphisms.
i(G) = sup?cT
i(G)  G ∈ G?. We close
Proposition 1. For template classes T ,T′, host classes G,G′and any host
graph G we have the following:
(i) cT
(ii) If T is closed under merging vertices within connected components then
cT
(iii) If T ⊆ T′then cT
i(G) for i = g,ℓ,f.
(iv) If G ⊆ G′then cT
(v) If¯G denotes the set of all subgraphs of G and we have T ∩¯G ⊆ T′∩¯G, then
cT
i(G) for i = g,ℓ.
(vi) If T and T′are closed under taking subgraphs, then (v) also holds for i = f.
g(G) ≥ cT
ℓ(G) ≥ cT
f(G)
ℓ(G) ≤ cT
f(G). In particular with (i) we conclude cT
i(G) ≥ cT′
i(G) ≤ cT
ℓ(G) = cT
f(G).
i(G′) for i = g,ℓ,f.
i(G) ≥ cT′
Example In order to examplify the notions introduced above consider the tem
plate class Cy of disjoint unions of cycles. As host graph G we take the Petersen
graph.
Page 6
Fig.1. From left to right: A global cover with three unions of cycles, a local cover of
size 5 with three cycles at each vertex, a folded cover with two preimages per vertex.
Note that the local cover does not yield a global cover.
Proposition 2. We have 3 = cCy
g (G) = cCy
ℓ(G) > cCy
f(G) = 2.
Proof. All witnesses for the upper bounds are shown in Figure 1. Clearly, cCy
2 since otherwise G would have to be a disjoint union of cycles. Now suppose,
cCy
ℓ(G) = 2. Since G is cubic, at each vertex there is exactly one edge contained
in two cycles of the covering. Thus, these edges form a perfect matching M of G.
Moreover, all cycles involved in the cover are alternating cycles with respect to
M. In particular they are all even. Since M is covered twice and the remaninig
edges of G once, the sum of sizes of cycles in the cover is 20. Hence a 6cycle C
must be involved. Now M restricted to G\V (C) must be perfect. But G\V (C)
is a claw.
f(G) ≥
⊓ ⊔
4 Template Classes
In this section we introduce the template classes and covering numbers corre
sponding to the columns of Table 1. We also include some known results and
general observations.
4.1Star Forests
The star arboricity sa(G) of a graph G, introduced by Akiyama and Kano [4], is
the minimum number of star forests, i.e., forests without paths of length 3, into
which the edgeset of G can be partitioned. In particular, if S denotes the class of
star forests, then sa(G) = cS
g(G). It is known that outerplanar and planar graphs
have star arboricity at most 3 and 5, respectively, see Hakimi et. al. [27]. That this
is best possible is shown by Algor and Alon [5]. Moreover, sa(G) ≤ tw(G)+1, see
Ding et. al. [15] and sa(G) ≤ 2deg(G) [27], where tw(G) and deg(G) denote the
treewidth (c.f. Section 5.2) and degeneracy (c.f. Section 5.1) of G, respectively.
Both bounds are bestpossible, see Alon et. al. [6] and Dujmovi´ c and Wood [16],
respectively.
Since merging any two vertices in a star and omitting double edges yields again
a star by Proposition 1 (ii) local and folded star arboricity coincide. In the
Page 7
following we relate the local star arboricity, denoted by saℓ(G), to two further
covering numbers. The arboricity a(G), introduced by NashWilliams [40], is the
minimum number of forests needed to cover the edges of G. It is known [40]
that a(G) = maxS⊆V (G)⌈E[S]
cycle per component and the pseudoarboricity p(G) is the minimum number of
pseudoforests needed to cover the edges of G. Picard and Queyranne [42] notice
that p(G) equals the minimum maximum outdegree of all orientations of G,
while the latter quantity equals maxS⊆V (G)⌈E[S]
Putting things together one obtains p(G) ≤ a(G) ≤ p(G) + 1. The local star
arboricity fits into the picture here.
S−1⌉. A pseudoforest is a graph with at most one
S⌉, see Frank and Gy´ arf´ as [21].
Theorem 3. For any graph G we have p(G) ≤ a(G) ≤ saℓ(G) ≤ p(G) + 1,
where either inequality can be strict. Moreover, saℓ(G) = p(G) if and only if G
has an orientation with maximum outdegree p(G) attained only at vertices of
degree p(G).
Proof. Every cover of G w.r.t. stars can be transferred into an orientation of G
by orienting every edge towards the center of the corresponding star. If every
vertex is contained in at most saℓ(G) stars, then the orientation has maximum
outdegree at most saℓ(G), i.e., p(G) ≤ saℓ(G).
In the same way every orientation can be transferred into a cover w.r.t. stars
by taking at every vertex the star of its incoming edges. If the orientation has
maximum outdegree p(G), then each vertex is contained no more than p(G)+1
stars, i.e., saℓ(G) ≤ p(G)+1. Moreover, the maximum outdegree is saℓ(G) if and
only if for every vertex v that is contained in saℓ(G) stars with centers different
from v there is no star with center v. Equivalently, saℓ(G) = p(G) if and only if
the maximum outdegree p(G) is attained only at vertices of degree p(G).
To prove a(G) ≤ saℓ(G) assume saℓ(G) = p(G). Otherwise (if saℓ(G) = p(G)+1)
the result follows from a(G) ≤ p(G) + 1. Hence, there is an orientation with
maximum outdegree p(G) attained only at vertices with degree p(G). Removing
these vertices we obtain a graph G′with p(G′) ≤ p(G)−1, in particular a(G′) ≤
p(G). We reinsert the vertices of degree p(G) putting each incident edge into a
different of the p(G) forests that partition G′. We obtain a cover of G with p(G)
forests, i.e., a(G) ≤ p(G) = saℓ(G).
Finally, we show that each inequality can be strict: First k = p(G) < a(G)
holds for every 2kregular graph. Secondly, we claim that k = p(G) = saℓ(G)
holds for the complete bipartite graph Kk,n with n =. Indeed, p(Kk,n) =
maxS⊆V (Kk,n)⌈E[S]
the bigger bipartition class yields saℓ(Kk,n) ≤ k.
It remains to present a graph G with k = a(G) < saℓ(G). It is known [15] that
a(G) ≤ tw(G). In Theorem 6 we show5that for every k there is a graph G with
tw(G) = k and i(G) ≥ k + 1. Then we have a(G) ≤ tw(G) = k < k + 1 ≤
i(G) ≤ tℓ(G) ≤ saℓ(G), where the nexttolast and last inequality follows by
Proposition 1 (i) and (iii), respectively.
S⌉ = ⌈kn
k+n⌉ = k and taking all maximal stars with centers in
⊓ ⊔
5i(G) and tℓ(G) are defined in Section 4.3.
Page 8
We will derive from Theorem 3 tight upper bounds for the local star arboricity
(c.f. Corollary 1 and 2), as well as a polynomial time algorithm to compute the
local star arboricity (c.f. Theorem 12).
4.2Caterpillar Forests
A graph parameter related to the star arboricity is the caterpillar arboricity
ca(G) of G. A caterpillar is a tree in which all nonleaf vertices form a path,
called the spline. The caterpillar arboricity is the minimum number of caterpillar
forests into which the edgeset of G can be partitioned. It has mainly been
considered for outerplanar graphs, see Kostochka and West [36], and for planar
graphs by Gon¸ calves and Ochem [22,23]. Since caterpillar forests are exactly
trianglefree interval graphs, ca(G) is related to the tracknumber of G defined
below.
4.3Interval Graphs
The class I of interval graphs has already been considered in many ways and
remains present in today’s literature. Interval graphs have been generalized to
intersection graphs of systems of intervals by several groups of people: Gy´ arf´ as
and West [26] propose the covering problem w.r.t. the template class I and
introduce the corresponding global covering number called the tracknumber,
denoted by t(G), i.e., t(G) = cI
g(G). It has been shown that outerplanar and
planar graphs have track number at most 2 [36] and 4 [23], respectively. Already
in 1979 Harary and Trotter [30] introduce the folded covering number w.r.t.
interval graphs, called the intervalnumber, denoted by i(G), i.e., i(G) = cI
It is known that trees have interval number at most 2 [30], outerplanar and planar
graphs have interval number at most 2 and 3, respectively, see Scheinermann and
West [45]. All these bounds are tight.
The local tracknumber tℓ(G) := cI
ℓ(G) is a natural variation of i(G) and t(G),
which to our knowledge has not been considered so far.
f(G).
5Results
In this section we present all the new results displayed in Table 1. We proceed
host class by host class.
5.1 Bounded Degeneracy
The degeneracy deg(G) of a graph G is the minimum of the maximum out
degree over all acyclic orientations of G. It is one less than the coloring number,
introduced in [18], and is a classical measure for the sparsity of G. The next
corollary follows directly from the definition of degeneracy and Theorem 3.
Corollary 1. For every G we have saℓ(G) ≤ deg(G) + 1.
Page 9
Let I be the class of interval graphs and Ca be the class of caterpillar forests, i.e.,
the class of bipartite interval graphs. Then by Proposition 1 (v) and (vi) we have
cI
i(G) for i = g,ℓ,f and every bipartite graph G. In particular, if G is
bipartite then t(G) = ca(G) and i(G) = caf(G). In Theorems 4, 6 and 7 below we
present graphs with high (folded) caterpillar arboricity. Since all these graphs are
bipartite, we obtain lower bounds on the tracknumber and intervalnumber of
those graphs. Indeed we always define a supergraph G of the complete bipartite
graph Km,n, for which track and intervalnumber has already been determined:
t(Km,n) = ca(Km,n) = ⌈
After a few definitions we present Lemma 1, which is applied in all our lower
bound results.
We refer to the bipartition classes in Km,n of size m and n by A and B, re
spectively. We need one more definition. For a cover ϕ of G w.r.t. T and a
subgraph H of G we define the restriction of ϕ to H as the cover ψ of H given
by ψ := ϕϕ−1(H). If T is closed under taking subgraphs, then ψ is also a cover
w.r.t. T .
i(G) = cCa
mn
m+n−1⌉ [26] and i(Km,n) = caf(Km,n) = ⌈mn+1
m+n⌉ [30].
Lemma 1. Let G be a graph with an induced Km,n, ϕ be a cover of G w.r.t. Ca
with s = max{ϕ−1(a) : a ∈ A}, and ψ be the restriction of ϕ to the subgraph G′
of G after removing all edges in Km,n. Then there are at least n − 2sm vertices
b ∈ B such that ψ−1(b) ≤ ϕ−1(b) − m.
Proof. Every vertex in a caterpillar C has at most 2 neighbors on the spline of
C. In particular, if ϕ : C1· ∪... · ∪Ck→ G and a ∈ A, then at most 2s incident
edges at a are covered by spline edges of some Ci. In other words there are at
least n − 2s vertices b ∈ B such that the edge {a,b} is covered under ϕ by a
nonspline edge with b being a leaf. Thus, for at least n − 2sm vertices b ∈ B
this is the case w.r.t. to every a ∈ A.
Now if {a,b} is covered by some edge e in Ci with b being a leaf, then in the
restriction of ϕ to G \ {a,b} the number of preimages of b is one less than in ϕ.
This concludes the proof.
⊓ ⊔
Theorem 4. For every k ≥ 1 there is a bipartite graph G such that 2deg(G) ≤
2k ≤ ca(G) = t(G).
Proof. The graph G consists of an induced Kk,n with A = k and B = n,
where n > (k −1)?2k−1
k−1
?+2k(2k−1), and for every ksubset S of B an induced
Kk,(k−1)2+1with smaller and larger bipartition class S and BS, respectively.
Orienting edges from B to A and from BSto S for every S proves deg(G) ≤ k.
Now consider an injective cover ϕ : C1· ∪... · ∪Csof G w.r.t. Ca and its restriction
ψ to the subgraph of G after removing all edges in Kk,n. Assume for the sake
of contradiction that the size s of ϕ is at most 2k − 1, i.e., max{ϕ−1(v) : v ∈
V (G)} ≤ s ≤ 2k − 1. Then by Lemma 1 there is a set W ⊂ B of at least
n − 2(2k − 1)k > (k − 1)?2k−1
k−1
?
vertices, such that ψ−1(b) ≤ ϕ−1(b) − k ≤
s−k ≤ k−1 for every b ∈ W. In particular, every b ∈ W has a preimage under ψ
in at most k−1 of the 2k−1 caterpillar forests. Since W > (k−1)?2k−1
k−1
?there
Page 10
is a kset S in W whose preimages are contained in at most k − 1 caterpillar
forests.
In other words, ψ restricted to G[S ∪ BS] is an injective cover of Kk,(k−1)2+1
w.r.t. Ca with size at most k − 1, which is impossible as ca(Kk,(k−1)2+1) =
⌈k(k−1)2+k
k+(k−1)2⌉ = k, due to [26].
⊓ ⊔
5.2Bounded (Simple) Treewidth
A ktree is a graph that can be constructed starting with a (k +1)clique and in
every step attaching a new vertex to a kclique of the already constructed graph.
We use the term stacking for this kind of attaching. The treewidth tw(G) of a
graph G is the minimum k such that G is a partial ktree, i.e., G is a subgraph
of some ktree [44].
We consider a variation of treewidth, called simple treewidth. A simple ktree
is a ktree with the extra requirement that there is a construction sequence in
which no two vertices are stacked onto the same kclique. (Equivalently, a ktree
is simple if it has a tree representation of width k in which every (k − 1)set of
subtrees intersects at most 2 treevertices.) Now, the simple treewidth stw(G) of
G is the minimum k such that G is a partial simple ktree, i.e., G is a subgraph
of some simple ktree.
For a graph G with stw(G) = k or tw(G) = k we fix any (simple) ktree that is a
supergraph of G and denote it by˜G. Clearly, G inherits a construction sequence
from˜G, where some edges are omitted.
Lemma 2. For every G we have tw(G) ≤ stw(G) ≤ tw(G) + 1.
Proof. The first inequality is clear. For the second inequality we show that every
ktree G is a subgraph of a simple (k +1)tree H. Whenever in the construction
sequence of G several vertices {v1,...,vn} are stacked onto the same kclique
C we consider C ∪ {v1} as a (k + 1)clique in the construction sequence for H.
Stacking now vionto C can be interpreted as stacking vionto C ∪ {vi−1} and
omitting the edge {vi−1,vi}. This way we can avoid multiple stackings onto k
cliques by considering (k + 1)cliques.
⊓ ⊔
Simple treewidth endows the notion of treewidth with a more topological flavor.
For a graph G we have the following: stw(G) ≤ 1 iff G is a linear forest, stw(G) ≤
2 iff G is outerplanar, stw(G) ≤ 3 iff G is planar and tw(G) ≤ 3 [17].
Simple treewidth also has connections to discrete geometry. In [11] a stacked
polytope is defined to be a polytope that admits a triangulation whose dual graph
is a tree. From that paper one easily deduces that a fulldimensional polytope
P ⊂ Rdis stacked if and only if stw(G(P)) ≤ d. Here G(P) denotes the 1skeleton
of P.
We consider both, graphs with bounded treewidth and graphs with bounded
simple treewidth, as host classes since A) most of the results for outerplanar
graphs are implied by the corresponding result for stw ≤ 2, B) lower bound
results for stw ≤ 3 carry over to planar graphs, C) our results differ for interval
Page 11
graphs as template class (c.f. Theorem 5 and 6), and D) when the maximum
covering numbers are the same for both classes, the lower bounds are slightly
stronger when provided by graphs of low simple treewidth.
Theorem 5. For every graph G we have tℓ(G) ≤ stw(G).
Proof. If stw(G) = 1, then G is a linear forest and hence an interval graph.
If stw(G) = 2, then G is outerplanar and even has tracknumber at most 2 as
shown in [36].
So let stw(G) = s ≥ 3. We build an injective cover ϕ : I1· ∪··· · ∪Ik → G with
ϕ−1(v) ≤ s for every v ∈ V (G) and Ii ∈ I for i ∈ [k]. We use as I1,...,Ik
only certain interval graphs, which we call slugs: A slug is like a caterpillar with
a fixed spline, except that the graph Iv
iinduced by the leaves at every spline
vertex v ∈ Iiis a linear forest6. The end vertices of a spline are called splineends
and vertices of degree at most 1 in Iv
iare called leafends; See Figure 2 for an
example. Note that slugs are indeed interval graphs.
v
Iv
i
e(C)
e(C)
e(C)
x
e(C) x
→
→
Fig.2. Left: A slug Ii with the spline drawn thick, splineends in white, and leafends
in grey. Right: How to extend the slug that contains the end e(C) by a new vertex x
when {w,w1} ∈ E(G) in Case 1 or {w,w2} ∈ E(G) in Case 2.
We define the cover ϕ along a construction sequence of G that is inherited
from a simple ktree˜G ⊇ G. At every step let H be the subgraph of G that is
already constructed (and hence already covered by ϕ). We maintain the following
invariants on ϕ, which allow us to stack a new vertex onto every kclique C onto
which no vertex has been stacked so far. We call such a clique stackable.
1) For every v ∈ V (H) there is a slug I(v), such that ϕ−1(v) contains a spline
vertex of I(v) and I(v) ?= I(w) for v ?= w.
2) For every stackable clique C there is a splineend or leafend e(C) of some
slug I with ϕ(e(C)) = w1∈ C, such that:
2a) If e(C) is a splineend, then I ?= I(v) for all v ∈ V (H).
2b) If e(C) is a leafend, then I = I(w2) for some w2∈ C \ {w1}.
2c) Every v is e(C) for at most two cliques C, and v = e(C) = e(C′) for
C ?= C′only if v is a splineend of degree 0 or a leafend of degree 1.
6In a caterpillar Iv
i is an independent set for every spline vertex v.
Page 12
It is not difficult to satisfy the above invariants for an initial kclique of˜G. Indeed,
this clique can be build up in a very similar way to the stacking procedure that
we describe now: In the construction sequence of G we are about to stack a
vertex w onto the stackable clique C = {w1,...,wk} of H, which means that for
every i ∈ [k] the clique (C \ {wi}) ∪ {w} in˜G is stackable in H ∪ {w}.
For i = 3,...,k we do the following. If {w,wi} ∈ E(G) we introduce a new leaf
x to I(wi) at the corresponding spline vertex in ϕ−1(wi), and if {w,wi} / ∈ E(G)
we introduce a new slug consisting only of x. Either way, we set ϕ(x) = w and
e((C\{wi−1})∪{w}) = x. For i = k we additionally set e((C\{wk})∪{w}) = x.
For w1and w2we distinguish two cases, which are illustrated in Figure 2:
In Case 1 e(C) is a splineend of some slug I. We first proceed with w2 as
with wi for i ≥ 3 above, i.e., e((C \ {w1}) ∪ {w}) is set to some new vertex
x. Now if {w,w1} ∈ E(G) we introduce a new spline vertex x to I adjacent to
e(C). If {w,w1} / ∈ E(G) we introduce a new slug I′consisting only of x. We
set ϕ(x) = w and I(w) = I if {w,w1} ∈ E(G) and I(w) = I′otherwise. Note
that 1) is satisfied since by 2a) I ?= I(v) for every vertex v in H.
In Case 2 e(C) is a leafend in the slug I(w2). We introduce a new slug I
consisting only of one vertex y and set ϕ(y) = w and I(w) = I. Now if {w,w2} / ∈
E(G) we introduce a new slug I′consisting only of x. If {w,w2} ∈ E(G) we
introduce a new leaf x to I(w2) at the same spline vertex as e(C). In case
{w,w1} ∈ E(G) the edge {y,e(C)} is added. Either way, we set ϕ(x) = w and
e((C \ {w1}) ∪ {w}) = x.
It is straightforward to check that ϕ is cover of H ∪ {w} w.r.t. I, and that ϕ
satisfies the invariants above. Note that since˜G is a simple ktree, the clique C is
no longer stackable and hence 2) need not be satisfied in H ∪{w}. Finally, every
stackable clique in H different from C was not affected by the above procedure,
which completes the proof.
⊓ ⊔
We can prove three lower bounds for covering numbers.
Theorem 6. For every k ≥ 1 there is a bipartite graph G such that stw(G) ≤
tw(G) + 1 ≤ k + 1 ≤ caf(G) = i(G).
Proof. Our graph G is Kk,nwith n = 2k2+ 1 where every vertex in the larger
bipartition class B has an additional private neighbor. It is easy to see that
tw(G) ≤ k and then Lemma 2 yields stw(G) ≤ tw(G) + 1.
Consider any cover ϕ of G w.r.t. Ca with s = max{ϕ−1(v) : v ∈ V (G)} and
its restriction ψ to the subgraph of G after removing all edges of Kk,n. By
Lemma 1 there are at least n − 2sk = 2k(k − s) + 1 vertices b ∈ B such that
ψ−1(b) ≤ ϕ−1(b)−k. Since b has a neighbor in G\Kk,nwe have ψ−1(b) ≥ 1
and hence s ≥ ϕ−1(b) ≥ k + 1, i.e., caf(G) ≥ k + 1.
⊓ ⊔
Theorem 7. For every k ≥ 3 there is a bipartite graph G such that stw(G)+1 ≤
k + 1 ≤ ca(G) = t(G).
Proof. The definition of the graph G starts with a Kk−1,m1with B = m1=
2(2k2− 2k + 1). We denote the vertices in B by {ui,vi: i ∈ [m1
2]}. For every
Page 13
i ∈ [m1
bipartition class Ai = {ui,vi} and Bi = {bij
every i ∈ [m1
2] and j ∈ [5] there is an induced Kk−1,m2with bipartition classes
Aij= {bij
Assume for the sake of contradiction that ϕ is an injective cover of G w.r.t.
Ca of size at most k. Consider the restriction ψ of ϕ to the subgraph G′=
G \ E(Kk−1,m1) of G. By Lemma 1 there are at least m1− 2k(k − 1) = 2k2−
2k + 2 >
2
vertices in b ∈ B with ψ−1(b) ≤ 1. In particular there is some
i ∈ [m1
by only one caterpillar forest, say ϕ−1(ui) ∈ Cu and ϕ−1(vi) ∈ Cv. Since G′
contains a 4cycle through uiand viwe have Cu?= Cv.
Now consider the restriction φ of ψ to the subgraph G′′= G′\ E[Ai∪ Bi] of
G′. Again by Lemma 1 there are at least 5(k − 1) − 4 vertices b ∈ Bi with
φ−1(b) ≤ k−2; more precisely Cu,Cv∩φ−1(b) = ∅. In particular there is some
j ∈ [5] such that Cu,Cv∩ φ−1(b) = ∅ for all b ∈ Aij.
In other words, φ restricted to G[Aij∪Bij] is an injective cover of Kk−1,(k−2)2+1
w.r.t. Ca with size at most k − 2, which is impossible as ca(Kk−1,(k−2)2+1) =
⌈(k−1)(k−2)2+k−1
k−1+(k−2)2
⌉ = k − 1, due to [26].
It remains to show that stw(G) ≤ k. Let A = {a1,...,ak−1} be the small
bipartition class of Kk−1,m1and Bij = {cij
Now construct G starting with A ∪ {u1,v1} and the following stackings (where
edges not in E(G) are omitted):
2] the graph G contains an induced K2,5(k−1)with smaller and larger
1,...,bij
k−1: j ∈ [5]}. Finally, for
1,...,bij
k−1} and Bijwith Bij = m2= (k − 2)2+ 1.
m1
2] such that ψ−1(ui),ψ−1(vi) ≤ 1, i.e., in G′each of ui,vi is covered
1,...,cij
m2} for all i ∈ [m1
2],j ∈ [5].
– Stack uionto A ∪ {vi−1} and vionto A ∪ {ui}
– Stack bi1
– Stack bij
∀i = 2,...,m1
∀i ∈ [m1
2],ℓ ∈ [k − 1]
2
ℓonto {a1,...,ak−ℓ−1,ui,vi,bi1
ℓonto {ui,vi,bi(j−1)
1
1,...,bi1
k−ℓ−1,bij
ℓ−1}
,...,bi(j−1)
1,...,bij
ℓ−1}
∀i ∈ [m1
ℓ−1} ∀i ∈ [m1
2],ℓ ∈ [k − 1],j ≥ 2
2],j ∈ [5],ℓ ≥ 2
– Stack cij
1onto Aij∪ {ui} and cij
ℓonto Aij∪ {cij
Since no kclique appears twice we conclude that stw(G) ≤ k.
⊓ ⊔
Theorem 8. For every k ≥ 2 there is a graph G such that
stw(G) + 1 ≤ k + 1 ≤ caf(G).
Proof. Fix k ≥ 2. We construct G starting with a star with k − 1 leaves
ℓ1,...,ℓk−1and center c1, seen as subgraph of a kclique. For n := 16k2−16k+4
and i = 2,...,n stack a new vertex cito ℓ1,...,ℓk−1,ci−1. Now stack vertices
s2,...,snto ℓ1,...,ℓk−2,ci−1,ci. Finally attach a leaf aito each of the si. This
may be viewed as stacking aito a subgraph of a kclique on ℓ1,...,ℓk−2,ci−1,si.
By construction stw(G) ≤ k.
Assume for the sake of contradiction that caf(G) ≤ k, i.e, there is a cover ϕ
of G w.r.t. Ca with ϕ−1(v) ≤ k for all v ∈ V (G). We consider three complete
bipartite edgedisjoint subgraphs H1,H2,H3of G induced by:
– A1:= ℓ1,...,ℓk−1and B1:= {c2i 1 ≤ i ≤ n/2}
– A2:= ℓ1,...,ℓk−1and B2:= {c2i−1 1 ≤ i ≤ n/2}
Page 14
– A3:= ℓ1,...,ℓk−2and B3:= {si 2 ≤ i ≤ n}
Note that Hiand Hj are edgedisjoint for i ?= j. Denote by ψ the restriction of
ϕ to G \ (E(H1) ∪ E(H2) ∪ E(H3)). We apply Lemma 1 three times, once for
each Hi, and obtain sets Wi⊂ Bi(i = 1,2,3) with W1,W2 ≥ n/2−2k(k−1)
and W3 ≥ n − 1 − 2k(k − 2), and ψ−1
(i = 1,2) and ψ−1
3(b) ≤ k − (k − 2) = 2 for b ∈ W3. From the choice of
n follows that there are consecutive ci,ci+1,ci+2,ci+3 ∈ W1∪ W2 such that
si+1,si+2,si+3 ∈ W3. Together with the leaves ai+1,ai+2,ai+3 these vertices
induce a 10vertex graph G′. It is not difficult to check that there is no cover
ψ of G′w.r.t Ca with ψ−1(ci+j) ≤ 1 for j = 0,1,2,3 and ψ−1(si+j) ≤ 2 for
j = 1,2,3 – a contradiction.
i(b) ≤ k − (k − 1) = 1 for b ∈ Wi
⊓ ⊔
5.3Planar and Outerplanar Graphs
Determining maximum covering numbers of (bipartite) planar graphs and out
erplanar graphs enjoys a certain popularity as demonstrated by the variety of
citations in Table 1. We add three new results to the list.
Theorem 9. The star arboricity of bipartite planar graphs is at most 4.
Proof. Every bipartite planar graph can be represented as the contact graph of
vertical and horizontal line segments in the plane [13]. Color the edges with four
colors depending on whether they are realized by an upper, lower, left, or right
endpoint of a segment. It is easy to see, that this yields a partition of the edges
into four star forests.
⊓ ⊔
Corollary 2. The local star arboricity of planar graphs and bipartite planar
graphs is at most 4 and 3, respectively.
Proof. As mentioned in Section 4.1 the arboricity a(G) of every graph G equals
maxS⊆V (G)⌈E[S]
planar bipartite graph has arboricity at most 3 and 2, respectively. Now the
statement follows from Theorem 3 and p(G) ≤ a(G).
S⌉ [40]. From this easily follows that every planar graph and
⊓ ⊔
The only question mark in Table 1 concerns the local tracknumber of planar
graphs. Scheinerman and West [45] show that the intervalnumber of planar
graphs is at most 3, but this is verified with a cover that is not injective. On the
other hand, there are bipartite planar graphs with tracknumber 4 [23]. However
by Theorem 9 and Theorem 5 every bipartite planar graph and every planar
graph of treewidth at most 3 has local tracknumber at most 3.
Conjecture 2. The local tracknumber of a planar graph is at most 3.
Page 15
6Complexity and Separability
In Table 1 we provide several pairs of a host class G and a template class T
for which the global covering number and the local covering number differ, i.e.,
cT
ℓ(G). Indeed this difference can be arbitrarily large.
g(G) > cT
Theorem 10. For the template class Cl of collections of cliques and the host
class G of line graphs, we have cCl
g(G) = ∞ and cCl
ℓ(G) ≤ 2.
Proof. By a result of Whitney [49] a graph G is a line graph iff cCl
To prove cCl
number of the line graph of the complete graph on n vertices is unbounded as n
goes to infinity. Assume L(Kn) is covered by k collections of cliques C1,...,Ck.
Every clique in L(Kn) corresponds in Kn to either a triangle or a star. We
disregard at most1
3n vertices of Knsuch that in the induced cover of the smaller
line graph no clique in C1corresponds to a triangle. Repeating this for every Ci,
we end up with a clique cover of L(Km) with m ≥ (2
cover of Kmwith star forests. Since the star arboricity of Kmis at leastm−1
we get k ≥m−1
2
> (2
ℓ(G) ≤ 2.
g(G) = ∞, we claim that cCl
g(L(Kn)) ∈ Ω(logn), i.e., the covering
3)kn that corresponds to a
2,
⊓ ⊔
3)k−1n, and thus k ∈ Ω(logn).
Remark 3. Milans [38] proved a similar result with interval graphs as template
class, i.e., t(L(Kn)) ∈ Ω(log∗(n)), while i(G) ≤ 2 for every line graph G.
Table 1 suggests that the separation of the local and the folded covering number
is more difficult. Indeed we have cT
except for the local tracknumber of planar graphs (c.f. Conjecture 2). However,
proving upper bounds for cT
ℓ(G) can be significantly more elaborate than for
cT
f(G), even if we suspect that both values are equal; see for example Conjecture 1
and Theorem 2.
ℓ(G) = cT
f(G) for every T and G in Table 1,
Observation 11 For the template class Cy of collections of cycles and the host
class G of paths, we have cCy
ℓ(G) = ∞ and cCy
f(G) ≤ 2.
Observation 11 may be considered pathological as there is no injective cover of
a path P w.r.t. a cycle and hence cCy
ℓ(P) = ∞. However, the local and folded
covering number may differ also if cT
ℓ(G) < ∞. We have provided one exam
ple for this when considering converings of the Petersen graph with disjoint
unions of cycles, see Proposition 2. There is another example: It is known that
i(Km,n) = ⌈mn+1
t(Km,n) presented in [14] indeed gives tℓ(Km,n) ≥ ⌈
tℓ(Km,n) > i(Km,n) for appropriate numbers m and n, e.g., n = m2− 2m + 2.
With Proposition 1 this translates into caℓ(Km,n) > caf(Km,n).
m+n⌉ [30] and t(Km,n) = ⌈
mn
m+n−1⌉ [26]. The lower bound on
mn
m+n−1⌉ and hence we have
Another interesting aspect concerns the computational complexity of the three
covering numbers. Very informally, one might suspect that the computation of
cT
f(G) is easier than of cT
ℓ(G), which in turn is easier than computing cT
g(G).