Induced and Weak Induced Arboricities

03/06/2018 ∙ by Maria Axenovich, et al. ∙ 0

We define the induced arboricity of a graph G, denoted by ia(G), as the smallest k such that the edges of G can be covered with k induced forests in G. This notion generalizes the classical notions of the arboricity and strong chromatic index. For a class F of graphs and a graph parameter p, let p(F) = {p(G) | G∈F}. We show that ia(F) is bounded from above by an absolute constant depending only on F, that is ia(F)≠∞ if and only if χ(F∇1/2) ≠∞, where F∇1/2 is the class of 1/2-shallow minors of graphs from F and χ is the chromatic number. Further, we give bounds on ia(F) when F is the class of planar graphs, the class of d-degenerate graphs, or the class of graphs having tree-width at most d. Specifically, we show that if F is the class of planar graphs, then 8 ≤ ia(F) ≤ 10. In addition, we establish similar results for so-called weak induced arboricities and star arboricities of classes of graphs.

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1 Introduction

For a graph , the arboricity is the smallest number of forests covering all the edges of . As Nash-Williams proved years ago, the arboricity is governed precisely by the largest density among the subgraphs of . Let

Theorem 1 (Nash-Williams [12]).

For every graph with we have .

This paper is concerned with the induced arboricity. Recall that a subgraph of a graph is induced if for any , if then . We define a subgraph of to be weak induced if each component of is an induced subgraph of , but itself is not neccessarily an induced subgraph of .

Definition 2.

The induced arboricity , respectively weak induced arboricity , is the smallest number of induced, respectively weak induced forests, covering .

For a complete graph , , we see that , since there could be at most one edge in any induced forest of , and because any weak induced forest of is a matching. Note that for the arboricity as well as the weak induced arboricity we can assume that an optimal set of forests covering the edges forms a partition of the edge set. For the induced arboricity the smallest number of induced forests covering the edges might be smaller than the smallest number of induced forests partitioning the edge set of a graph.

Proposition 3.

For each there is a graph with and an edge , such that in any cover of with induced forests, is contained in all of them.

The notion of induced arboricity has been considered only recently, see for example a paper by some of the authors of the current paper, [3]. However, a special case of this parameter, when induced forests are required to be induced matchings, is a classical parameter, the strong chromatic index. The strong chromatic index , introduced by Erdős and Nešetřil (c.f. [10]), is the smallest for which can be covered with induced matchings. In addition, the star arboricity was considered extensively, where is the smallest number of star-forests, that is, forests with each component being a star, needed to cover the edges of .

For a graph parameter and a class of graphs , let . We are concerned with induced arboricity and weak induced arboricity . In addition, we consider the induced star arboricity and weak induced star arboricity of a graph defined as the smallest number of induced star-forests and weak induced star-forests, respectively, covering the edges of . Note that for the induced star arboricity as well as the weak induced star arboricity we can assume that an optimal set of forests covering the edges forms a partition of the edge set.

Further note that every star-forest is a forest, an induced forest is also a weak induced forest, which in turn is also a forest. Every matching is a weak induced star-forest and every induced matching is an induced star-forest. Thus, denoting the edge-chromatic number of , we have

(1)
(2)
(3)
Proposition 4.

For any graph we have , , and .

This shows that star arboricities behave in a sense similar to their respective arboricities. In particular Proposition 4 implies that for any graph class we have

Next, we characterize exactly all graph classes for which and for which . The characterization for induced arboricity is in terms of so-called shallow minors, a notion introduced by Nešetřil and Ossona de Mendez [13]. A graph on vertex set is a -shallow minor of if there exist pairwise vertex-disjoint stars in with centers such that for each edge in there is an edge in between and , or between and . The set of all -shallow minors of is denoted and the set of all -shallow minors of graphs from a class is denoted .

Theorem 5.

For any graph class we have

The acyclic chromatic number, , of a graph is the smallest number of colors in a proper vertex-coloring such that the union of any two color classes forms an induced forest of . A result in [3] gives the following dependencies between induced arboricity and the acyclic chromatic number

(4)

Thus, for any class of graphs we have if and only if . In turn Dvořák [9] characterizes classes with in terms of shallow topological minors. For a graph let denote the graph obtained from by subdividing each edge exactly once. Let and for a family of graphs let . Note that and hence  [13]. Dvořák [9] proved that if and only if . This gives the following corollary of Theorem 5 which is of independent interest.

Corollary 6.

For any graph class we have

Theorem 5 shows that the weak induced arboricity is a parameter behaving in a sense like arboricity. However, induced arboricity is more complex and depends on the structure of the graph, not only on the density. In particular for the class of -degenerate graphs we give the maximum values for arboricities and weak arboricities and show that the induced arboricities are unbounded. For several classes of graphs with more restricted structure, that is, , we provide bounds on the various arboricities. Specifically we consider graphs of given acyclic chromatic number, given tree-width, and the class of all planar graphs.

The degeneracy of a graph is the smallest integer  such that every subgraph of has a vertex of degree at most . If the degeneracy of is , then  [5]. That is, the degeneracy is closely related to the arboricity and hence the density of a graph. For an integer , let denote the class of all graphs of degeneracy at most . Theorem 5 shows that , , , and are bounded while .

Theorem 7.

For the class of all graphs of degeneracy , , we have

Note that is a strengthening of the result from [2].

The tree-width of a graph which is the smallest integer  such that is a subgraph of some chordal graph of clique number  [14]. For a positive integer , let denote the class of all graphs of acyclic chromatic number at most  and let denote the class of all graphs of tree-width at most . It is well-known that for each , that is, graphs of tree-width  have acyclic chromatic number at most . Inequalities (4) state a functional dependency between induced arboricity and the acyclic chromatic number. Here we establish upper bounds and specific values for the other arboricities on . Further we show that for each these bounds are attained on the strict subclass , a result of independent interest.

Theorem 8.

Let , , and if is even and if

is odd. Then

Moreover, and .

Finally we consider the family of planar graphs. Note that  [4] and for any  [15].

Theorem 9.

For the class of all planar graphs we have

For the class of all outerplanar graphs, we have

The bounds are also proved in [7].

We prove Propositions 3 and 4 and give the main construction in Section 2. Theorems 59 are proved in the respective sections.

2 Preliminary results

Proof of Proposition 3

Let . Consider a complete bipartite graph with bipartition , , . Every induced forest in is a star. Hence any induced forest in contains at most edges, implying that . Moreover, an induced forest has edges if and only if contains exactly one vertex of and all vertices of , and taking one such forest for each vertex in gives the unique (up to isomorphism) cover of with induced forests. Thus . Note that every vertex in is contained in all forests.

Figure 1: Left: A graph with an edge and , where in every cover of with two induced forests is contained in both forests. Right: The unique (up to isomorphism) cover of with two induced forests.

Now let be the graph obtained from two vertex-disjoint copies of , together with an edge joining a vertex from the larger bipartition part of with a vertex from the larger bipartition part of . See Figure 1 for an illustration. Taking induced forests in covering and induced forests in covering , we can pair these up and add to each of them to get induced forests in covering . Thus . Observe that this gives the unique (up to isomorphism) cover of with induced forests, and that is contained in all such forests.∎

Proof of Proposition 4

For a set of forests, and each forest , fix a root of each tree in arbitrarily and let be a subgraph of formes by the edges at distance  from the respective root in . We refer to ’s as layers of . Then we see that each layer forms a star-forest and is the union of all ’s.

Consider to be a smallest set of forests (respectively weak induced forests) covering . The forests formed by the even-indexed layers or by the odd-indexed layers , , form star-forests (respectively weak induced star-forests) covering . This implies that and .

Let be a smallest set of induced forests covering . Consider each third layer of each forest and their unions: , , , . These forests form induced star-forests covering . This implies that .∎

2.1 Construction of the graph

For each we shall construct a graph which is used in the proofs later. Let be the -vertex tree in the left part of Figure 2. For we shall define as follows. Let be the power of a path on vertices, that is, and all pairs of vertices at distance at most  in are edges in . Split the vertex set of into pairwise disjoint sets of consecutive vertices each. Let be the family of these sets. For each , let , where consists of the first vertices and the vertex in according to the order in and . Note that each of and induces a clique on vertices in . Add new vertices, denoted , , , and add all edges between and and between and for all ’s. Call the resulting graph . Finally, let be obtained as a union of and a set of pairwise vertex-disjoint cliques that each clique shares exactly one vertex with . That is, we “hang” ’s on the vertices of and refer to them as “hanging cliques”.

Figure 2: Two graphs of tree-width with and . Left: and has tree-width . Right: and has tree-width .
Figure 3: Illustration of the planar graph with and . Left: The graph is the fourth power of a -vertex path (bold edges). The triangles induced by , for are highlighted in gray and form a facial triangle in . Right: is obtained from by augmenting each set , with a vertex respectively and four “hanging ’s”.

See the right part of the Figure 2 for an illustrative example when and Figure 3 for the case .

Lemma 10.

For any , the tree-width of is , , and . Moreover is outerplanar and is planar.

Proof.

First observe that has tree-width  and satisfies , , and .

Consider and let . As is a chordal graph of clique number , the tree-width of is . Moreover, is the union of and a family of edge-disjoint copies of , each of which is chordal, has tree-width , and shares at most  vertices with . It follows that also is chordal and has clique number . Therefore the tree-width of is .

Next we show that . To this end, consider a smallest set of forests of induced stars (not necessarily induced forests) partitioning the edge set of . Without loss of generality we assume that each edge is contained in exactly one forest in . If a star has only two vertices, we arbitrarily pick one vertex to be the center and the other to be the only leaf. We orient each edge in from the leaf to the center of the star containing . Thus, is at least the out-degree of any vertex. The number of edges in the subgraph of is . Each of these edges contributes  to the out-degree of some vertex in . Thus, there is a set or (assume without loss of generality that it is ) of total out-degree at least in . Then there is a vertex with out-degree at least in . Thus there is a set of at least  weak induced star-forests such that in each of the forests in the vertex is a leaf of some star whose center is in .

On the other hand, consider the edges of incident to that are in the “hanging clique”, that is, that are not in . These edges are in distinct forests from and neither of them is in , since such a forest would contain a whose endpoint is a center of a star. Thus is in at least forests. This proves that .

To prove that consider any smallest set of induced star-forests covering the edge set of . Without loss of generality we assume that each edge is contained in exactly one forest in . Again, we orient each edge in from the leaf to the center of the star containing . As argued above, without loss of generality, there is a set of total out-degree at least in . By the choice of , at least stars in have a leaf in and the center in . At most such stars have their center in because there are edges in . Thus there is a set of at least stars in that have a leaf in and the center in . Since induces a clique, the stars from belong to distinct forests of .

Let . Each vertex in is incident to edges of a “hanging clique” which belong to distinct forests of because any two such edges either induce a triangle in the respective “hanging clique” or have an edge of between their endpoints. Call the set of these forests . We see that and are disjoint. Thus .

The facts that is outerplanar and is planar follow from the embeddings given in Figure 2 and Figure 3, respectively. ∎

3 Proof of Theorem 5

First suppose that . We shall prove that . We have and since , . Thus due to Dvořák [9]. Hence as for any graph by inequality (4).

Now suppose that . Then holds due to the following claim.

Claim 1.

For every graph we have .

Let be a proper vertex coloring of with at most colors. This exists as and hence is -degenerate. Moreover, let , be induced star-forests covering , which exist as .

Now consider with vertex set to be an arbitrary but fixed -shallow minor of . Let be defined by vertex-disjoint stars in with centers , respectively. Let , where contains all edges for which and contains all remaining edges for which is adjacent to a leaf in , or is adjacent to a leaf in .

We define a vertex coloring of as follows: For let be the indices of those star-forests that contain at least one edge in . For each vertex define the color of to be . Clearly, uses at most colors.

Figure 4: Two stars and (black edges) with centers and , an edge (dashed) between a leaf of and the center of . If an induced subgraph (bold gray edges) contains and an edge in , then it also contains .

If is an edge in , then and hence . Assume that is an edge in , say center is adjacent to a leaf in , consider the index such that edge is in induced star-forest .

Then since otherwise all the edges , and some edge from are induced by in and contradict that is an induced star-forest. See Figure 4 for an illustration. Therefore and thus . So is indeed a proper vertex coloring of the -shallow minor of , proving that , which concludes the proof of Claim 1.

Next we prove that if and only if . For each graph we have by inequality (1). Thus implies . So assume that . Then holds due to the follow claim.

Claim 2.

For every graph we have .

Let be a smallest set of star-forests covering and let be an optimal proper coloring of . For all and let denote the subgraph of formed by all edges whose leaf vertex is colored under . Then is a weak induced star-forest. This shows that . Here by Proposition 4 and as each graph is -degenerate [5]. This concludes the proof of Claim 2 and the proof of Theorem 5.

4 Proof of Theorem 7

Let . The fact that follows from Nash-Williams’ Theorem. For the lower bound it suffices to observe that the complete bipartite graph , with satisfies and by Nash-Williams’ Theorem.

Some of the authors of this paper constructed -degenerate graphs of arbitrarily large induced arboricity, see [3], showing that . We shall show first that , for any -degenerate graph .

An ordering of the vertices of is a -degeneracy ordering if for each the vertex has at most  neighbors with . We think of identifying vertices with some points on a horizontal line. We say that a vertex lies to the right (left) of if (). A star is a right (left) star with respect to the vertex ordering if its center lies left of (right of) all its leaves. We shall prove the following claim by induction on .

Claim.

For any -degenerate graph and any -degeneracy ordering of , there is a coloring of in colors and sets of colors , , such that each color class is a weak induced star-forest of right stars, , and does not contain any color from for and , i.e., of the edges going to the left from .

When , let . All conditions are clearly satisfied. Assume that and consider a -degeneracy ordering of . Let . Then is a -degeneracy ordering of . Consider a coloring of guaranteed by the induction hypothesis and respective color sets . Let be the neighbors of in with , . Further let , i.e., the sets are obtained from by deleting the colors of the edges going from to the right, to another neighbor of . Then for each . Now we shall extend the coloring to a coloring of , such that , , and so on, , . This procedure is possible since , . Let such that . We shall prove that this coloring and the color sets satisfy the Claim.

Within each color class is a union of right stars by induction. Moreover all monochromatic (left) stars centered at are single edges whose color is in , that is, their color is not assigned to any edge going from to the left. Hence all color classes are unions of right stars. The sets satisfy the conditions of the Claim. We only need to verify that each color class is a weak induced star-forest. Since each color class in is a weak induced star-forest, it is sufficient to consider monochromatic stars containing an edge for some . The color of the edge is in . Thus neither of the neighbors of with is a leaf in the monochromatic (right) star containing . Hence this star is induced in . This proves the Claim.

The claim in particular shows that . Next, we shall show that for every there is a -degenerate bipartite graph with .

Let be fixed and let . Consider pairwise disjoint sets , , and , for each , where and , . Let be a union of complete bipartite graphs with parts and , . It is easy to see that the resulting graph is -degenerate and bipartite. So assume that is covered by some number of weak induced forests. Without loss of generality assume that each edge is contained in exactly one of these forests. Consider a coloring such that if is in the forest. Let . For each vertex , let . We see that there are such tuples , . The total possible number of such distinct tuples is . Thus, by pigeonhole principle, there is a set such that for any two elements , we have . This implies that under coloring the subgraph is a union of monochromatic stars, each with center in and leaf-set . Since each color class is a forest, all these stars have different colors. So in total there are at least  colors. By a similar argument, we see that there is a subset of so that , is a union of monochromatic stars with centers in and leaf-sets . Note that a star with center and leaf-set and a star with center and leaf-set together do not induce a forest, so they must be of different colors. Thus the total number of colors is at least , a contradiction. Thus . This concludes the proof of Theorem 7.

5 Proof of Theorem 8

First we shall establish the upper bounds. Recall that . So let be an -vertex graph. First we consider the arboricity of . Consider an acyclic coloring of with colors and let be the sizes of the color classes. Then and since each edge is induced by some two color classes and each pair of color classes induces a forest, we have that

Since each subgraph of on vertices has acyclic chromatic number at most , it has at most edges. Thus and by Nash-Williams’ Theorem (Theorem 1) for any .

The upper bound on the star arboricity follows from a result of Hakimi et al. [11] showing that . Hence for any .

Next we shall show that . Consider an acyclic coloring of with color classes . Consider a complete graph on vertex set and a partition of into matchings , where . For let be the spanning subgraph of consisting of those edges with , , and . It is easy to see that is a partition of into weak induced forests, which implies that . Thus .

We have by inequality (4). The remaining upper bounds follow from Proposition 4, which gives and . Moreover follows from Theorem 7 since each graph in is -degenerate.

To see the lower bounds, note first that the complete graph has tree-width , , and . Moreover Dujmović and Wood [8] show that and