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Domination When the Stars Are Out∗ Domination when the Stars Are Out∗ Danny Hermelin Matthias Mnich Erik Jan van Leeuwen Gerhard J. Woeginger January 16, 2019 Abstract We algorithmize the structural characterization for claw-free graphs by Chudnovsky and Seymour. Building on this result, we show that Dominating Set on claw-free graphs is (i) fixed-parameter tractable and (ii) even possesses a polynomial kernel. To complement these results, we establish that Dominating Set is unlikely to be fixed-parameter tractable on the slightly larger class of graphs that exclude K1;4 as an induced subgraph (K1;4-free graphs). We show that our algorithmization can also be used to show that the related Connected Dominating Set problem is fixed-parameter tractable on claw-free graphs. To complement that result, we show that Connected Dominating Set is unlikely to have a polynomial kernel on claw-free graphs and is unlikely to be fixed-parameter tractable on K1;4-free graphs. Combined, our results provide a dichotomy for Dominating Set and Connected Dominat- ing Set on K1;`-free graphs and show that the problem is fixed-parameter tractable if and only if ` ≤ 3. 1 Introduction (Connected) Dominating Set is the problem to determine whether a given graph G has a (connected) dominating set of size at most k. A subset D ⊆ V (G) is a dominating set if every ver- tex in G is either contained in D or adjacent to some vertex in D, and D is a connected dominating set if D is a dominating set and G[D] (the graph induced by D) is connected. Dominating sets play a prominent role in both algorithmics and combinatorics (see e.g. [46, 47]). Since the (Con- nected) Dominating Set problem is hard in its decision [54, 41], approximation [34, 43], and parameterized versions [26], research has focused on finding graph classes for which the problem becomes tractable. In this paper, we focus on the class of claw-free graphs. A graph is claw-free if no vertex has three pairwise nonadjacent neighbors, i.e. if it does not contain K1;3 as an induced subgraph. The class of claw-free graphs contains several well-studied graph classes, including line graphs, arXiv:1012.0012v3 [cs.DS] 14 Jan 2019 unit interval graphs, complements of triangle-free graphs, and graphs of several polyhedra and polytopes. Throughout the years, this graph class attracted much interest, and is by now the subject of hundreds of mathematical research papers; for an overview we refer to the survey by Faudree et al. [33]. In the context of algorithms, most research on claw-free graphs has focussed on the Indepen- dent Set problem. Building on Edmond's classical polynomial-time algorithm [29] for Inde- pendent Set on line graphs (better known as Matching), Sbihi [77] and Minty [67] (the latter ∗An extended abstract of this paper appeared in the proceedings of ICALP 2011, LNCS vol. 6755, Springer, Berlin, 2011, pp. 462{473. This work is supported by the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7/2007-2013) under REA grant agreement number 631163.11, by the Israel Science Foundation (grant no. 551145/14), by the European Research Council under ERC Starting Grant 306465 (BeyondWorstCase), and by DFG grant MN 59/1-1. 1 corrected by Nakamura and Tamura [70]) already gave polynomial-time algorithms for Indepen- dent Set on claw-free graphs over 30 years ago. Recently, significantly faster algorithms have been discovered that follow a similar approach [31, 72]. In contrast to the Independent Set problem, however, Dominating Set is NP-complete on claw-free graphs. In fact, Dominating Set is NP-complete even on line graphs [82]. Nevertheless, Fernau [36] recently showed that Dominating Set on line graphs (also known as Edge Domi- nating Set) has an f(k) · nO(1) time algorithm, where k is the size of the solution, meaning that this problem is fixed-parameter tractable (see e.g. [19, 27, 28, 38]). Moreover, Fernau [36] showed that any instance of Dominating Set on line graphs can be reduced in polynomial time to have O(k2) vertices, which implies that the problem admits a polynomial kernel (see e.g. [19, 27, 28, 38]). Both results were recently slightly improved [81, 52]. It has been left an open question, however, whether such algorithms also exist for Dominating Set on claw-free graphs. In a wider picture, claw-free graphs are a member of the more general family of graphs that exclude K1;` as an induced subgraph for ` 2 N, i.e. `-claw-free graphs. These graphs generalize many important classes of geometric intersection graphs; for example, unit square graphs are K1;5-free, and unit disk graphs are K1;6-free. Marx showed that Dominating Set is W[1]- hard on unit square graphs when parameterized by the solution size k, implying it is W[1]-hard on K1;5-free graphs [66], which makes it unlikely that the problem is fixed-parameter tractable (see e.g. [19, 27, 28, 38]). However, the problem becomes trivial on K1;2-free graphs, since these graphs are just disjoint unions of cliques. Hence, the computational and parameterized complexity of Dominating Set has been left open on K1;3-free (claw-free) and K1;4-free graphs. In this paper, we resolve the question whether Dominating Set is fixed-parameter tractable on K1;3-free and/or K1;4-free graphs. We answer this same question also for the related Connected Dominating Set problem. Additionally, we determine the `-claw-free graphs on which these problems admit a polynomial kernel. These results combined completely settle the parameterized complexity of both problems on `-claw-free graphs. Our Results. In the first part of the paper, we present our main contribution: an algorith- mic version of a recent, highly nontrivial structural decomposition theorem for claw-free graphs by Chudnovsky and Seymour. This decomposition shows that every claw-free graph can be built by applying certain gluing operations to certain atomic structures. The proof that such a decomposition exists is contained in a sequence of seven papers by Chudnovsky and Sey- mour [10, 11, 12, 13, 14, 15, 16]; an accessible summary of the proof can be found in the an- nouncement of their results [9]. The original proof of their decomposition theorem, however, does not directly imply an algorithm to find the decomposition. In order to obtain our algorithmic decomposition theorem for claw-free graphs, we first develop polynomial-time algorithms that undo the aforementioned gluing operations (Sect. 3). These algorithms could be of independent interest, as the structures that these algorithms find are not specific to claw-free graphs. Then, we develop polynomial-time algorithms that recognize several of the atomic structures that make up claw-free graphs (Sect. 4). Finally, we make several changes to the proof of the original decomposition theorem by Chudnovsky and Seymour that simplify it and make it easier to algorithmize (Sect. 5). Finally, we put all these pieces together to give an algorithmic decomposition theorem for claw-free graphs that runs in O(n2m3=2) time (Sect. 6). The structural result for claw-free graphs that is implied by our algorithmic decomposition theorem is inspired by a claim of Chudnovsky and Seymour [9, Claim 3.1] in the announcement of their structural result for claw-free graphs. However, as far as we are aware, this claim is not explicitly proved in their final work [10, 11, 12, 13, 14, 15, 16]. Our algorithmic decomposition theorem can be seen as a variant or an interpretation of [9, Claim 3.1] with the hindsight of knowing the final work of Chudnovsky and Seymour, as well as an explicit proof and an algorithm to actually find the decomposition. In the second part of the paper, we apply our algorithmic decomposition theorem for claw-free graphs to establish the following: 2 • Dominating Set on claw-free graphs is fixed-parameter tractable when parameterized by the solution size. To be precise, we show that we can decide the existence of a dominating set of size at most k in 9k · O(n5) time when the graph is claw-free (Sect. 7). • Connected Dominating Set on claw-free graphs is fixed-parameter tractable when pa- rameterized by the solution size. To be precise, we show that we can decide the existence of a connected dominating set of size at most k in 36k · nO(1) time when the graph is claw-free (Sect. 8). This resolves an open question by Misra et al. [68]. • Dominating Set on claw-free graphs has a polynomial kernel when parameterized by the solution size. To be precise, we show that given a claw-free graph G and an integer k, we 3 can output a graph G0 with O(k ) vertices and an integer k0 such that G has a dominating set of size k if and only if G0 has a dominating set of size k0; the presented algorithm runs in O(n5) time (Sect. 9). In the third part of the paper (Sect. 10), we complement the above results and show that: • Dominating Set and Connected Dominating Set are W[1]-hard on K1;4-free graphs. • Connected Dominating Set has no polynomial kernel on claw-free graphs (even on line graphs), unless the polynomial hierarchy collapses to the third level. • Dominating Set and Connected Dominating Set on claw-free graphs (even on line graphs) cannot have an algorithm that runs in 2o(k)nO(1) time, where k is the size of the solution, unless the Exponential Time Hypothesis fails.
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