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Planarization and Acyclic Colorings of Subcubic Claw-Free Graphs

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Planarization and Acyclic Colorings of Subcubic Claw-Free Graphs Planarization and acyclic colorings of subcubic claw-free graphs Christine Cheng!, Eric McDermid!!, and Ichiro Suzuki!!! Department of Computer Science, University of Wisconsin–Milwaukee, Milwaukee, WI 53211, USA ccheng, mcdermid, suzuki @uwm.edu { } Abstract. We study methods of planarizing and acyclically coloring claw-free subcubic graphs. We give a polynomial-time algorithm that, given such a graph G, produces an independent set Q of at most n/6 vertices whose removal from G leaves an induced planar subgraph P (in fact, P has treewidth at most four). We further show the stronger result that in polynomial-time a set of at most n/6 edges can be identified whose removal leaves a planar subgraph (of treewidth at most four). From an approximability point of view, we show that our results imply 6/5- and 9/8-approximation algorithms, respectively, for the (NP-hard) problems of finding a maximum induced planar subgraph and a maximum planar subgraph of a subcubic claw-free graph, respectively. Regarding acyclic colorings, we give a polynomial-time algorithm that finds an optimal acyclic vertex coloring of a subcubic claw-free graph. To our knowledge, this represents the largest known subclass of subcu- bic graphs such that an optimal acyclic vertex coloring can be found in polynomial-time. We show that this bound is tight by proving that the problem is NP-hard for cubic line graphs (and therefore, claw-free graphs) of maximum degree d 4. An interesting corollary to the algo- ≥ rithm that we present is that there are exactly three subcubic claw-free graphs that require four colors to be acyclically colored. For all other such graphs, three colors suffice. 1 Introduction A simple, finite graph G is said to be claw-free if no vertex of G has three pairwise nonadjacent neighbors. It is subcubic if every vertex of G has degree at most three. Claw-free graphs are a well-studied and interesting class of graphs that generalize line graphs. Additionally, claw-free graphs are very well-understood, thanks to a complete structure theorem found by Chudnovsky and Seymour [6]. In this paper we explore methods of planarization and acyclic colorings of claw-free subcubic graphs. ! Supported by NSF award CCF-0830678. !! Supported by NSF award CCF-0830678 and UWM Research Growth Initiative. !!! Supported by UWM Research Growth Initiative. 2 Planarization. Planarization is a broad term used to refer to the process of modifying a graph in order to make it planar (see [14] for a survey). Interest in such techniques arises both for combinatorial and theoretical uses, but also for in- dustrial applications (for example, VLSI and layout problems). In the minimum nonplanar vertex deletion problem (MINVD) we wish to compute a smallest set of vertices Q such that G Q is an induced planar subgraph. The complement problem of MINVD is to find− a maximum induced planar subgraph (MAXIPS), where the size of the solution is the number of vertices of G Q. Analogously, the minimum nonplanar edge deletion problem MINED asks− for a smallest set of edges E! such that G E! is a planar subgraph. The complement of MINED is to find a maximum planar− subgraph (MAXPS), where the size of the solution is the number of edges of G E!. − From a complexity perspective, all of these optimization problems are hard, but some positive results are also known. On the negative side, Yannakakis [18, 19] first showed that MINVD and MINED are NP-hard. Independently, Liu and Geldmacher [15] also proved that MINED is NP-hard. Cˇalinescu et al. [4] improved upon this by showing that MINED and MAXPS are Max SNP-hard. Subsequently, Faria et al. [9–11] (note the two different sets of authors) showed that MINVD and MINED are both Max SNP-hard in cubic graphs, and that MAXIPS and MAXPS are NP-hard in cubic graphs. They also showed that MAXIPS in general graphs admits no polynomial-time approximation algorithm with a fixed ratio unless P = NP. We remark that, although Faria et al. [11] do not explicitly state it, the derived instance in their hardness proof for MAXIPS in cubic graphs has the property that every vertex is in a triangle (a necessary and sufficient condition for claw-freeness in a cubic graph). Hence MAXIPS is NP-hard in claw-free cubic graphs. It is also easy to see that their hardness result for MAXPS in cubic graphs implies that MAXPS is also NP-hard in claw-free cubic graphs: a cubic graph G contains a planar subgraph P of size n k (i.e., − we remove k edges to obtain P ) if and only if the claw-free graph G! obtained by replacing each vertex of G with a triangle has a planar subgraph of size 3n k. − On the positive side, Faria et al. [11] gave a 4/3-approximation algorithm for MAXIPS in subcubic graphs. Edwards and Farr [7] presented a polynomial- time algorithm that, given a graph of maximum degree d, finds an induced planar subgraph of size at least 3n/(d + 1). They later generalized this result to graphs of average degree d (in fact, the induced subgraph found is series- parallel). This can be interpreted as a (d + 1)/3-approximation algorithm, which matches the bound given by Faria et al. when the maximum degree is three. Finally, Cˇalinescu et al. [4] gave a 9/4-approximation for MAXPS in general graphs. We remark that there is much other work on finding subgraphs (induced or otherwise) having a particular property – for example, Cˇalinescu et al. [5] have recently given an approximation for maximum series-parallel subgraphs. Also, different algorithmic approaches have been taken, such as the FPT algorithm of Marx and Schlotter [16] for MINVD. Acyclic colorings. An acyclic coloring of a graph G is a proper vertex coloring of G with the property that no cycle is bicolored. The acyclic coloring problem 3 is to compute an acyclic coloring for G using the fewest number of colors. Aside from its intrinsic interest, researchers are interested in this problem because it has applications in computing the Jacobian and Hessian of sparse matrices [12]. Using probabilistic methods, Alon et al. [1] proved that it is always possible to acyclically color the vertices of a graph of maximum degree d using O(d4/3) colors, and showed that there exists graphs requiring Ω(d4/3/(log d)1/3) colors. For fixed values of d, they showed that a straightforward greedy algorithm uses at most d2 + 1 colors. Several authors have presented polynomial-time algorithms [3, 13, 17] showing that graphs of maximum degree 3, 4, and 5, respectively, admit an acyclic 4-, 5-, and 7-coloring. Note that all of these results are approximation algorithms – they will not, in general, return an acyclic coloring with the fewest number of colors possible. Regarding special cases of graphs, Zhang and Bilka [20] (implicitly) give a polynomial-time algorithm showing that every cubic line graph admits an acyclic coloring using at most three colors – a result that will be useful to us later on. We remark that much attention has also been devoted to the acyclic edge coloring of a graph G (defined analogously to acyclic vertex colorings). For our purposes, we mention only that Alon and Zaks [2] have shown that determining if a cubic graph G admits an acyclic coloring with at most three colors is NP- complete. Our Contribution. On the surface, planarization and acyclic colorings seem to be quite different problems. A unifying approach to our results is our sim- ple reduction technique, that, in linear time, yields a reduced graph with very desirable properties. In particular, we show that if the reduced graph admits a (induced) planar subgraph of a certain size, or is acyclically colorable, then the original graph also has this property. We believe that this reduction technique could be applicable to other problems restricted to subcubic claw-free graphs. Regarding planarization, we give a polynomial-time algorithm that, given a claw-free subcubic graph, produces an independent set Q of at most n/6 ver- tices whose removal from G leaves an induced planar subgraph P (in fact, P has treewidth at most four). Hence, G always has an induced planar subgraph (of treewidth at most four) of size at least 5n/6. Given this result, we further show the stronger result that in polynomial-time a set of at most n/6 edges can be identified whose removal leaves a planar subgraph (of treewidth at most four). From the perspective of the combinatorial bounds given by Edwards and Farr [8], our results show the existence of significantly larger induced planar subgraphs when restricted to subcubic claw-free graphs, rather than general subcubic graphs. From an approximability point of view, we show that our first two bounds give a 6/5- and a 9/8-approximation ratio, respectively, for MAX- IPS and MAXPS. This gives a better performance guarantee for the special case of subcubic claw-free graphs than that of the 4/3-approximation algorithm for MAXIPS in general subcubic graphs, and an improvement for this special case over the general 9/4-approximation algorithm for MAXPS. Next, we give a polynomial-time algorithm that finds an optimal acyclic vertex coloring of a subcubic claw-free graph. To our knowledge, this represents 4 Fig. 1. A claw with base u, a diamond and double diamond with endpoints u and v, and an envelope with an acyclic coloring. the largest subclass of subcubic graphs (or any non-trivially degree bounded class of graphs) such that an optimal acyclic vertex coloring can be found in polynomial-time.
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