A complexity dichotomy and a new boundary class for the dominating set problem

We study the computational complexity of the dominating set problem for hereditary graph classes, i.e., classes of simple unlabeled graphs closed under deletion of vertices. Every hereditary class can be defined by a set of its forbidden induced subgraphs. There are numerous open cases for the compl...

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Published in:	Journal of combinatorial optimization Vol. 32; no. 1; pp. 226 - 243
Main Author:	Malyshev, D. S.
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.07.2016
Subjects:	Combinatorics Convex and Discrete Geometry Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Theory of Computation Boundary class Polynomial-time algorithm Dominating set Computational complexity
ISSN:	1382-6905, 1573-2886
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Abstract	We study the computational complexity of the dominating set problem for hereditary graph classes, i.e., classes of simple unlabeled graphs closed under deletion of vertices. Every hereditary class can be defined by a set of its forbidden induced subgraphs. There are numerous open cases for the complexity of the problem even for hereditary classes with small forbidden structures. We completely determine the complexity of the problem for classes defined by forbidding a five-vertex path and any set of fragments with at most five vertices. Additionally, we also prove polynomial-time solvability of the problem for some two classes of a similar type. The notion of a boundary class is a helpful tool for analyzing the computational complexity of graph problems in the family of hereditary classes. Three boundary classes were known for the dominating set problem prior to this paper. We present a new boundary class for it.
AbstractList	We study the computational complexity of the dominating set problem for hereditary graph classes, i.e., classes of simple unlabeled graphs closed under deletion of vertices. Every hereditary class can be defined by a set of its forbidden induced subgraphs. There are numerous open cases for the complexity of the problem even for hereditary classes with small forbidden structures. We completely determine the complexity of the problem for classes defined by forbidding a five-vertex path and any set of fragments with at most five vertices. Additionally, we also prove polynomial-time solvability of the problem for some two classes of a similar type. The notion of a boundary class is a helpful tool for analyzing the computational complexity of graph problems in the family of hereditary classes. Three boundary classes were known for the dominating set problem prior to this paper. We present a new boundary class for it.
Author	Malyshev, D. S.
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References	AbouEisha H, Hussain S, Lozin V, Monnot J, Ries B (2014) A dichotomy for upper domination in monogenic classes. In: Combinatorial optimization and applications. Lecture notes in computer science, vol 8881. Springer, pp 258–267 AlekseevVKorobitsynDLozinVBoundary classes of graphs for the dominating set problemDiscret Math20042851–316207483510.1016/j.disc.2004.04.0101121.05081 ClarkBColbournCJohnsonDUnit disk graphsAnn Discret Math199048165177108856910.1016/S0167-5060(08)71047-10739.05079 AlekseevVA polynomial algorithm for finding the largest independent sets in fork-free graphsDiskretn Analiz i Issled Oper Ser 11999643190931.05078[in russian] LozinVMoscaRIndependent sets in extensions of 2K2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2K_2$$\end{document}-free graphsDiscret Appl Math20041461 7480211223710.1016/j.dam.2004.07.0061087.90080 BrandstädtADraganFLeH-OMoscaRNew graph classes of bounded clique-widthTheory Comput Syst2005385623645215693910.1007/s00224-004-1154-61084.68088 MalyshevDThe complexity of the edge 3-colorability problem for graphs without two induced fragments each on at most six verticesSib Electron Math Rep2014118118221326.05047 KratschDDomination and total domination in asteroidal triple-free graphsDiscret Appl Math2000991–3111123174382710.1016/S0166-218X(99)00128-60943.05063 Lokshtantov D, Vatshelle M, Villanger Y (2014) Independent set in P5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_5$$\end{document}-free graphs in polynomial time. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, pp 570–581 GolovachPPaulusmaDList coloring in the absence of two subgraphsDiscret Appl Math2014166123130316316610.1016/j.dam.2013.10.0101283.05096 KorobitsynDOn the complexity of domination number determination in monogenic classes of graphsDiscret Math Appl1992221911991084070 MalyshevDClasses of graphs critical for the edge list-ranking problemJ Appl Ind Math201482 245255318526410.1134/S1990478914020112 LozinVBoundary classes of planar graphsComb Probab Comput2008172287295239635410.1017/S09635483070088141166.05016 BroersmaHGolovachPPaulusmaDSongJUpdating the complexity status of coloring graphs without a fixed induced linear forestTheor Comput Sci20124141919289657910.1016/j.tcs.2011.10.0051234.68129 Graphs: 5-vertices. Information system on graph classes and their inclusions. http://www.graphclasses.org/smallgraphs.html#nodes5. Accessed 27 Jan 2015 Graphclass: AT-free. Information system on graph classes and their inclusions. http://www.graphclasses.org/classes/gc_61.html. Accessed 27 Jan 2015 Malyshev D (2013) The complexity of the 3-colorability problem in the absence of a pair of small forbidden induced subgraphs. Discret Math (submitted) BertossiADominating sets for split and bipartite graphsInform Process Lett1984191374076162310.1016/0020-0190(84)90126-10539.68058 AlekseevVOn easy and hard hereditary classes of graphs with respect to the independent set problemDiscret Appl Math20031321–31726202426110.1016/S0166-218X(03)00387-11029.05140 BacsóGTuzaZDominating cliques in P5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_5$$\end{document}-free graphsPeriod Math Hung19902130330810.1007/BF023526940746.05065 ZverovichIThe domination number of (Kp,P5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(K_p, P_5)$$\end{document}-free graphsAustralas J Comb2003279510019553901022.05060 Alekseev V (1983) On the local restrictions effect on the complexity of finding the graph independence number. In: Combinatorial-algebraic methods in applied mathematics. Gorkiy University Press, Gorkiy, pp 3–13 [in russian] GolovachPPaulusmaDRiesBColoring graphs characterized by a forbidden subgraphDiscret Appl Math20151801101110328069810.1016/j.dam.2014.08.0081303.05061 Schweitzer P (2014) Towards an isomorphism dichotomy for hereditary graph classes. arXiv, arXiv:1411.1977 YannakakisMGavrilFEdge dominating sets in graphsSIAM J Appl Math198038336437257942410.1137/01380300455.05047 CourcelleBMakowskyJRoticsULinear time optimization problems on graphs of bounded clique widthTheory Comput Syst2000332125150173964410.1007/s0022499100091009.68102 Problem: Domination. Information system on graph classes and their inclusions. http://www.graphclasses.org/classes/problem_Domination.html. Accessed 27 Jan 2015 Kral’DKratochvilJTuzaZWoegingerGComplexity of coloring graphs without forbidden induced subgraphsLect. Notes Comput Sci20012204254262190563710.1007/3-540-45477-2_231042.68639 GolovachPPaulusmaDSongJ4-coloring H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H$$\end{document}-free graphs when H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H$$\end{document} is smallDiscret Appl Math20131611–2140150297335710.1016/j.dam.2012.08.0221259.05061 AlekseevVBoliacRKorobitsynDLozinVNP-hard graph problems and boundary classes of graphsTheor Comput Sci20073891–2219236236337410.1016/j.tcs.2007.09.0131143.68058 Lozin V, Malyshev D (2015) Vertex coloring of graphs with few obstructions. Discret Appl Math. doi:10.1016/j.dam.2015.02.015 GareyMJohnsonDComputers and intractability: a guide to the theory of NP-completeness1979New YorkFreeman0411.680391979 KratschSSchweitzerPGraph isomorphism for graph classes characterized by two forbidden induced subgraphsLect Notes Comput Sci201275513445304012510.1007/978-3-642-34611-8_706102415 D Malyshev (9872_CR29) 2014; 8 M Yannakakis (9872_CR32) 1980; 38 V Lozin (9872_CR24) 2008; 17 P Golovach (9872_CR14) 2014; 166 9872_CR30 9872_CR31 G Bacsó (9872_CR7) 1990; 21 9872_CR2 9872_CR1 9872_CR18 9872_CR17 B Courcelle (9872_CR12) 2000; 33 A Brandstädt (9872_CR9) 2005; 38 D Malyshev (9872_CR28) 2014; 11 I Zverovich (9872_CR33) 2003; 27 D Kratsch (9872_CR21) 2000; 99 B Clark (9872_CR11) 1990; 48 V Lozin (9872_CR26) 2004; 146 V Alekseev (9872_CR3) 1999; 6 V Alekseev (9872_CR5) 2007; 389 M Garey (9872_CR13) 1979 P Golovach (9872_CR16) 2013; 161 V Alekseev (9872_CR4) 2003; 132 V Alekseev (9872_CR6) 2004; 285 9872_CR23 9872_CR25 H Broersma (9872_CR10) 2012; 414 P Golovach (9872_CR15) 2015; 180 S Kratsch (9872_CR22) 2012; 7551 9872_CR27 D Kral (9872_CR20) 2001; 2204 D Korobitsyn (9872_CR19) 1992; 2 A Bertossi (9872_CR8) 1984; 19
References_xml	– reference: BrandstädtADraganFLeH-OMoscaRNew graph classes of bounded clique-widthTheory Comput Syst2005385623645215693910.1007/s00224-004-1154-61084.68088 – reference: BroersmaHGolovachPPaulusmaDSongJUpdating the complexity status of coloring graphs without a fixed induced linear forestTheor Comput Sci20124141919289657910.1016/j.tcs.2011.10.0051234.68129 – reference: Graphs: 5-vertices. Information system on graph classes and their inclusions. http://www.graphclasses.org/smallgraphs.html#nodes5. Accessed 27 Jan 2015 – reference: KratschSSchweitzerPGraph isomorphism for graph classes characterized by two forbidden induced subgraphsLect Notes Comput Sci201275513445304012510.1007/978-3-642-34611-8_706102415 – reference: ClarkBColbournCJohnsonDUnit disk graphsAnn Discret Math199048165177108856910.1016/S0167-5060(08)71047-10739.05079 – reference: Alekseev V (1983) On the local restrictions effect on the complexity of finding the graph independence number. In: Combinatorial-algebraic methods in applied mathematics. Gorkiy University Press, Gorkiy, pp 3–13 [in russian] – reference: YannakakisMGavrilFEdge dominating sets in graphsSIAM J Appl Math198038336437257942410.1137/01380300455.05047 – reference: Schweitzer P (2014) Towards an isomorphism dichotomy for hereditary graph classes. arXiv, arXiv:1411.1977 – reference: Kral’DKratochvilJTuzaZWoegingerGComplexity of coloring graphs without forbidden induced subgraphsLect. Notes Comput Sci20012204254262190563710.1007/3-540-45477-2_231042.68639 – reference: AbouEisha H, Hussain S, Lozin V, Monnot J, Ries B (2014) A dichotomy for upper domination in monogenic classes. In: Combinatorial optimization and applications. Lecture notes in computer science, vol 8881. Springer, pp 258–267 – reference: AlekseevVOn easy and hard hereditary classes of graphs with respect to the independent set problemDiscret Appl Math20031321–31726202426110.1016/S0166-218X(03)00387-11029.05140 – reference: KorobitsynDOn the complexity of domination number determination in monogenic classes of graphsDiscret Math Appl1992221911991084070 – reference: Malyshev D (2013) The complexity of the 3-colorability problem in the absence of a pair of small forbidden induced subgraphs. Discret Math (submitted) – reference: GolovachPPaulusmaDRiesBColoring graphs characterized by a forbidden subgraphDiscret Appl Math20151801101110328069810.1016/j.dam.2014.08.0081303.05061 – reference: LozinVBoundary classes of planar graphsComb Probab Comput2008172287295239635410.1017/S09635483070088141166.05016 – reference: GolovachPPaulusmaDList coloring in the absence of two subgraphsDiscret Appl Math2014166123130316316610.1016/j.dam.2013.10.0101283.05096 – reference: Graphclass: AT-free. Information system on graph classes and their inclusions. http://www.graphclasses.org/classes/gc_61.html. Accessed 27 Jan 2015 – reference: BertossiADominating sets for split and bipartite graphsInform Process Lett1984191374076162310.1016/0020-0190(84)90126-10539.68058 – reference: Lokshtantov D, Vatshelle M, Villanger Y (2014) Independent set in P5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_5$$\end{document}-free graphs in polynomial time. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, pp 570–581 – reference: LozinVMoscaRIndependent sets in extensions of 2K2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2K_2$$\end{document}-free graphsDiscret Appl Math20041461 7480211223710.1016/j.dam.2004.07.0061087.90080 – reference: GareyMJohnsonDComputers and intractability: a guide to the theory of NP-completeness1979New YorkFreeman0411.680391979 – reference: MalyshevDThe complexity of the edge 3-colorability problem for graphs without two induced fragments each on at most six verticesSib Electron Math Rep2014118118221326.05047 – reference: GolovachPPaulusmaDSongJ4-coloring H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H$$\end{document}-free graphs when H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H$$\end{document} is smallDiscret Appl Math20131611–2140150297335710.1016/j.dam.2012.08.0221259.05061 – reference: KratschDDomination and total domination in asteroidal triple-free graphsDiscret Appl Math2000991–3111123174382710.1016/S0166-218X(99)00128-60943.05063 – reference: ZverovichIThe domination number of (Kp,P5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(K_p, P_5)$$\end{document}-free graphsAustralas J Comb2003279510019553901022.05060 – reference: Lozin V, Malyshev D (2015) Vertex coloring of graphs with few obstructions. Discret Appl Math. doi:10.1016/j.dam.2015.02.015 – reference: AlekseevVA polynomial algorithm for finding the largest independent sets in fork-free graphsDiskretn Analiz i Issled Oper Ser 11999643190931.05078[in russian] – reference: AlekseevVKorobitsynDLozinVBoundary classes of graphs for the dominating set problemDiscret Math20042851–316207483510.1016/j.disc.2004.04.0101121.05081 – reference: CourcelleBMakowskyJRoticsULinear time optimization problems on graphs of bounded clique widthTheory Comput Syst2000332125150173964410.1007/s0022499100091009.68102 – reference: AlekseevVBoliacRKorobitsynDLozinVNP-hard graph problems and boundary classes of graphsTheor Comput Sci20073891–2219236236337410.1016/j.tcs.2007.09.0131143.68058 – reference: Problem: Domination. Information system on graph classes and their inclusions. http://www.graphclasses.org/classes/problem_Domination.html. Accessed 27 Jan 2015 – reference: BacsóGTuzaZDominating cliques in P5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_5$$\end{document}-free graphsPeriod Math Hung19902130330810.1007/BF023526940746.05065 – reference: MalyshevDClasses of graphs critical for the edge list-ranking problemJ Appl Ind Math201482 245255318526410.1134/S1990478914020112 – volume: 11 start-page: 811 year: 2014 ident: 9872_CR28 publication-title: Sib Electron Math Rep – volume: 285 start-page: 1 issue: 1–3 year: 2004 ident: 9872_CR6 publication-title: Discret Math doi: 10.1016/j.disc.2004.04.010 – volume: 389 start-page: 219 issue: 1–2 year: 2007 ident: 9872_CR5 publication-title: Theor Comput Sci doi: 10.1016/j.tcs.2007.09.013 – volume: 38 start-page: 623 issue: 5 year: 2005 ident: 9872_CR9 publication-title: Theory Comput Syst doi: 10.1007/s00224-004-1154-6 – volume: 48 start-page: 165 year: 1990 ident: 9872_CR11 publication-title: Ann Discret Math doi: 10.1016/S0167-5060(08)71047-1 – volume: 33 start-page: 125 issue: 2 year: 2000 ident: 9872_CR12 publication-title: Theory Comput Syst doi: 10.1007/s002249910009 – ident: 9872_CR30 – volume: 414 start-page: 9 issue: 1 year: 2012 ident: 9872_CR10 publication-title: Theor Comput Sci doi: 10.1016/j.tcs.2011.10.005 – volume: 146 start-page: 74 issue: 1 year: 2004 ident: 9872_CR26 publication-title: Discret Appl Math doi: 10.1016/j.dam.2004.07.006 – volume: 132 start-page: 17 issue: 1–3 year: 2003 ident: 9872_CR4 publication-title: Discret Appl Math doi: 10.1016/S0166-218X(03)00387-1 – volume: 99 start-page: 111 issue: 1–3 year: 2000 ident: 9872_CR21 publication-title: Discret Appl Math doi: 10.1016/S0166-218X(99)00128-6 – ident: 9872_CR18 – volume: 19 start-page: 37 issue: 1 year: 1984 ident: 9872_CR8 publication-title: Inform Process Lett doi: 10.1016/0020-0190(84)90126-1 – ident: 9872_CR2 – ident: 9872_CR25 doi: 10.1016/j.dam.2015.02.015 – ident: 9872_CR31 – volume: 6 start-page: 3 issue: 4 year: 1999 ident: 9872_CR3 publication-title: Diskretn Analiz i Issled Oper Ser 1 – volume: 7551 start-page: 34 year: 2012 ident: 9872_CR22 publication-title: Lect Notes Comput Sci doi: 10.1007/978-3-642-34611-8_7 – volume: 2204 start-page: 254 year: 2001 ident: 9872_CR20 publication-title: Lect. 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Snippet	We study the computational complexity of the dominating set problem for hereditary graph classes, i.e., classes of simple unlabeled graphs closed under...
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SubjectTerms	Combinatorics Convex and Discrete Geometry Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Theory of Computation
Title	A complexity dichotomy and a new boundary class for the dominating set problem
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