Domination when the stars are out
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Domination when the stars are out
Domination when the Stars Are OutDanny HermeliiT Matthias Mnicl?Erik .Jan van Loeuwen*Gerhard .J. Woeginger5arXiv: 1012.0012vl [cs.DS] 30 Nov 2010Abst Domination when the stars are out tractWe algorithmize the recent structural characterization for claw-free graphs by Chuduovsky 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 Domination when the stars are out that Dominating Set is not fixed-parameter tractable on the slightly larger class of graphs t hat, exclude Ki'i as an induced subgraph. Our results pDomination when the stars are out
rovide a dichotmny for Dominating Set in A*|.rfree graphs ami show that the problem is fixed-parameter tractable if and only if f. < 3. Finally, we shDomination when the Stars Are OutDanny HermeliiT Matthias Mnicl?Erik .Jan van Loeuwen*Gerhard .J. Woeginger5arXiv: 1012.0012vl [cs.DS] 30 Nov 2010Abst Domination when the stars are out s.’Max-Planck-Institut fill Infill matik. Saarbnicken, Germany, hernelinCttpi-inf.npg.de.’International Computer Science Institute. Berkeley, USA, nnnichCic8i.berkeley.edu.’Department of Informatics, University of Bergen, Norway, E.J.van.LeeuvenCii.uib.no.3 Department of Mathematics and Computer Sci Domination when the stars are out ence, TU Eindhoven, The Netherlands, guoegiCwin.tue.nl1https://khothuvien.cori!1IntroductionThe dominating set problem is the problem of determining wDomination when the stars are out
het her a given graph G has a dominating set of a specified size k. (A subset L) c V(G) is dominat ing if every vertex in G is either contained in D oDomination when the Stars Are OutDanny HermeliiT Matthias Mnicl?Erik .Jan van Loeuwen*Gerhard .J. Woeginger5arXiv: 1012.0012vl [cs.DS] 30 Nov 2010Abst Domination when the stars are out ent irely to dominating sets, while Hcdctnicmi and Laskar list over 300 papers related to domination in graphs [29]. In complexity theory. Dominating Set was one of the first problems recognized as NP-complctc (32), and one of t he first examples of an NP-hard optimizat ion problem with an approxim Domination when the stars are out ation algorithm [31 whose approximation factor guarantee is tight under reasonable assumptions (21). Dominating Set also plays a central role in paramDomination when the stars are out
eterized complexity. It motivated the definition of the W-hicrarchy. being the first natural example of a problem complete for the class w[2], and hasDomination when the Stars Are OutDanny HermeliiT Matthias Mnicl?Erik .Jan van Loeuwen*Gerhard .J. Woeginger5arXiv: 1012.0012vl [cs.DS] 30 Nov 2010Abst Domination when the stars are out vable by an algorithm with running time f(k) ■ nO[i' for any computable function f (15).Since the dominating set problem is hard in its decision, approximation, and parameterized versions, research has focused on finding special graph classes for which the problem becomes t ract able. In this paper Domination when the stars are out we consider the class of claw-free graphs. A graph is claw-free if no vertex has t hree pairwise nonadjacent neighbors, i.c. if it docs not contain /<Domination when the stars are out
1.3 as an induced subgraph. The class of claw-free graphs contains several well-studied graph classes as a special case, including line graphs, unit iDomination when the Stars Are OutDanny HermeliiT Matthias Mnicl?Erik .Jan van Loeuwen*Gerhard .J. Woeginger5arXiv: 1012.0012vl [cs.DS] 30 Nov 2010Abst Domination when the stars are out cted much interest., and is by now t he subject of hundreds of mathematical research papers and surveys [19].In the context of algorithms, initial study was direct cd towards extending algorithms dcvelojied for line graphs. The foremost examples arc the two independent results by Shibi [41] and Mint Domination when the stars are out y [35] (t he latt er corrected by Nakamura and Tamura [37]), which extend Edmond’s classical polynomialtime algorithm for Maximum Independent Set on lDomination when the stars are out
ine graphs 17). better known as the maximum matching, to t he class of claw-free graphs. In contrast to the independent set problem. Dominat ing set oDomination when the Stars Are OutDanny HermeliiT Matthias Mnicl?Erik .Jan van Loeuwen*Gerhard .J. Woeginger5arXiv: 1012.0012vl [cs.DS] 30 Nov 2010Abst Domination when the stars are out ithm, where k is the of the solution [23]. Whether this result extends to claw-free graphs has boon left, an open question.There has also been considerable study devoted to graphs excluding / 3. Those ty|>es of graphs appoiu- frequently when considering geometric intersection graphs of Domination when the stars are out various types. For instance, unit, square graphs are A'|,5-frce. and unit, disk graphs are Ki.fl-frec. Marx showed that Dominating Set is w[l]-hard onDomination when the stars are out
unit squares graphs, implying that it. is also hard on A'l^-free graphs [34]. Note that the problem becomes easy on A'lj-frec graphs, since these graDomination when the Stars Are OutDanny HermeliiT Matthias Mnicl?Erik .Jan van Loeuwen*Gerhard .J. Woeginger5arXiv: 1012.0012vl [cs.DS] 30 Nov 2010Abst Domination when the stars are out clude these graphs as an induced subgraph and not as a subgraph. In the latter case. Dominating Set Is actually known to be fixed-parameter tractable due to a result by Philip Ct al. [38].1.1Our results In this paper, wo show how to extend the algorithm of Fermin [23] from line graphs to daw-free gr Domination when the stars are out aphs. For this, we use a recent, highly nontrivial structural cliaracierization of claw-free graphs due to Chudnovsky and Seymour. This characterize iDomination when the stars are out
on shows that every claw-free graph can be built by applying certain gluing operations to certain atomic structures. The proof of t his characterizat Domination when the Stars Are OutDanny HermeliiT Matthias Mnicl?Erik .Jan van Loeuwen*Gerhard .J. Woeginger5arXiv: 1012.0012vl [cs.DS] 30 Nov 2010Abst Domination when the stars are out n ideas in the proof.The original proof of the Chudnovsky and Seymour characterization theorem for claw-free1graphs is essentially nonalgorithmic. Thus, the main challenge in our approach was to provide an algorithmic- version of this theorem. We fully meet and settle this challenge in Scx’tion 2.Se Domination when the stars are out veral attempts to characterize claw free graphs and its subclasses have been made before the full decomposition theorem of Chudnovsky and Seymour appeDomination when the stars are out
ared. For instance, Fouquct [24] showed that certain claw hoc graphs arc either so called quasi line graphs or possesses a vertex with a C-, as an indDomination when the Stars Are OutDanny HermeliiT Matthias Mnicl?Erik .Jan van Loeuwen*Gerhard .J. Woeginger5arXiv: 1012.0012vl [cs.DS] 30 Nov 2010Abst Domination when the stars are out ntly developed a now polynomial lime algorithm for Maximum Independent Sot on claw free graphs. For Dominating Set, however, the core difficulty of the problem persists even in these quasi line graphs. Hence such a much finer decomposition Is needed, which wo develop in this paper. Moreover, we beli Domination when the stars are out eve that having such a fine decomposition will prove useful in attacking other optimization problems on claw-free graphs.Using oin algorithmic claw-frDomination when the stars are out
ee decomposition, we establish the following:•Dominating Set on claw-free graphs is fixed-parameter tractable. To be precise, we show that we can deciDomination when the Stars Are OutDanny HermeliiT Matthias Mnicl?Erik .Jan van Loeuwen*Gerhard .J. Woeginger5arXiv: 1012.0012vl [cs.DS] 30 Nov 2010Abst Domination when the stars are out ractable and solvable in O‘(36fc) t ime (Section 3). This resolves an open question due to Misra et al. [36].•Dominating Set on claw-free graphs Iras a polynomial kernel with O(À'*) vertices (Section 5).To complement our results, wo show that Dominating Set is vv[l]-hard on A|,4-froc graphs (sec Sec Domination when the stars are out tion 6 for the proof). Tints, we completely determine the parameterized complexity stat ILS of Dominating Set in Aiy-frec graphs lor all values of2AllDomination when the stars are out
Algorithmic View of the Structure of Claw-Free GraphsWe give algorithms to find t he decomposition of claw-free graphs implied by t he structural chaDomination when the Stars Are OutDanny HermeliiT Matthias Mnicl?Erik .Jan van Loeuwen*Gerhard .J. Woeginger5arXiv: 1012.0012vl [cs.DS] 30 Nov 2010Abst Domination when the stars are out that the size of the maximum independent set of the graph is greater than tlnce. Due to space limitations, we only state this theorem here and defer its full proof to the appendix.2.1Basic Definit ions 'lb understand the st ructure theorem, we need to introduce a significant amount of notation. All Domination when the stars are out notions and dcfinit.ioiLs an? essentially the same as ill [10].We work wit h a more general type of graph, a so-called t rigraph. A bigtnph is a grapDomination when the stars are out
h wit h a distinguished subset of edges, that are calk’d Sf^tii-exlyr.s and form a matching. Two vertices are called Hc-miadjaceut if t here is a semiDomination when the Stars Are OutDanny HermeliiT Matthias Mnicl?Erik .Jan van Loeuwen*Gerhard .J. Woeginger5arXiv: 1012.0012vl [cs.DS] 30 Nov 2010Abst Domination when the stars are out them. A graph is then simply a t.rigraph that has no semi-edges. We now say that II, V are adjacent if II, V are either st rongly adjacent or seiniadjacent, and u, V are anliadjaccnl if 11, V are eit her strongly antiadjacent or semiadjacent. In a similar manner, we can distinguish (strong) neighbo Domination when the stars are out rhoods, completeness, cliques, simplicial vertices, etc., as well as (strong) anti-neighborhDomination when the stars are out
) to denote t he maximum size of a stable set of G’.A trigraph G’ is a thickcniiuj of a trigraph G’ if for every V c ViG’*) there is a nonempty set x„Gọi ngay
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