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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.tu-berlin.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

interval-number, track-number, and linear-, star-, and caterpillar arboric-

ity. As host graphs we consider graphs of bounded degeneracy, bounded

degree, or bounded (simple) tree-width, 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 graph-theoretical papers at all, Petersen shows that any

2r-regular 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 FE-340/8-1 as part of ESF project GraDR EU-

ROGIGA

⋆⋆Research was supported by GraDR EUROGIGA project No. GIG/11/E023

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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 track-number [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 coloring-aspect 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 interval-number [30], in the latter

case as the splitting-number [32] of a graph.

While some covering numbers, like arboricities, are of mainly theoretical interest,

others, like thickness, interval-number and track-number 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) tree-width, 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.

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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 non-edge 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 best-known 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].

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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 edge-disjoint path. As before, a set W1,...,Wkof

walks covers G, denoted by G =?

of the edge-sets 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 interval-number

i(G)). It is a weakening of LLAC above.

i∈[k]Liwith every

i∈[k]Wi, if the edge-set 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 track-number 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 edge-surjective 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 ϕ|Ti,

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∈V|ϕ−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 track-number,

respectively.

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cT

f(G) = min?maxv∈V|ϕ−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 n-vertex 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.

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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 6-cycle 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 edge-set 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

tree-width (c.f. Section 5.2) and degeneracy (c.f. Section 5.1) of G, respectively.

Both bounds are best-possible, 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

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following we relate the local star arboricity, denoted by saℓ(G), to two further

covering numbers. The arboricity a(G), introduced by Nash-Williams [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 out-degree 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 out-degree 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

out-degree 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 out-degree p(G), then each vertex is contained no more than p(G)+1

stars, i.e., saℓ(G) ≤ p(G)+1. Moreover, the maximum out-degree 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 out-degree 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 out-degree 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 2k-regular 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 next-to-last 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.

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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 non-leaf vertices form a path,

called the spline. The caterpillar arboricity is the minimum number of caterpillar

forests into which the edge-set 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

triangle-free interval graphs, ca(G) is related to the track-number 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 track-number,

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 interval-number, 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 track-number 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.

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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 track-number and interval-number of

those graphs. Indeed we always define a super-graph G of the complete bipartite

graph Km,n, for which track- and interval-number 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 := ϕ|ϕ−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

non-spline 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 k-subset 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

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is a k-set 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) Tree-width

A k-tree is a graph that can be constructed starting with a (k +1)-clique and in

every step attaching a new vertex to a k-clique of the already constructed graph.

We use the term stacking for this kind of attaching. The tree-width tw(G) of a

graph G is the minimum k such that G is a partial k-tree, i.e., G is a subgraph

of some k-tree [44].

We consider a variation of tree-width, called simple tree-width. A simple k-tree

is a k-tree with the extra requirement that there is a construction sequence in

which no two vertices are stacked onto the same k-clique. (Equivalently, a k-tree

is simple if it has a tree representation of width k in which every (k − 1)-set of

subtrees intersects at most 2 tree-vertices.) Now, the simple tree-width stw(G) of

G is the minimum k such that G is a partial simple k-tree, i.e., G is a subgraph

of some simple k-tree.

For a graph G with stw(G) = k or tw(G) = k we fix any (simple) k-tree 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

k-tree 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 k-clique

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 tree-width endows the notion of tree-width 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 tree-width 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 full-dimensional polytope

P ⊂ Rdis stacked if and only if stw(G(P)) ≤ d. Here G(P) denotes the 1-skeleton

of P.

We consider both, graphs with bounded tree-width and graphs with bounded

simple tree-width, 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 tree-width.

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 track-number 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 spline-ends

and vertices of degree at most 1 in Iv

iare called leaf-ends; 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, spline-ends in white, and leaf-ends

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 k-tree˜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 k-clique 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 spline-end or leaf-end e(C) of some

slug I with ϕ(e(C)) = w1∈ C, such that:

2a) If e(C) is a spline-end, then I ?= I(v) for all v ∈ V (H).

2b) If e(C) is a leaf-end, 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 spline-end of degree 0 or a leaf-end 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 k-clique 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 spline-end 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 leaf-end 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 k-tree, 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 4-cycle 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 k-clique 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 k-clique. 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 k-clique 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 edge-disjoint 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 edge-disjoint 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 10-vertex 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

end-point 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 track-number of planar

graphs. Scheinerman and West [45] show that the interval-number 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 track-number 4 [23]. However

by Theorem 9 and Theorem 5 every bipartite planar graph and every planar

graph of treewidth at most 3 has local track-number at most 3.

Conjecture 2. The local track-number 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 track-number 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).