Graphs with maximal induced matchings of the same size

Abstract A graph is well-indumatched if all its maximal induced matchings are of the same size. We first prove that recognizing whether a graph is well-indumatched is a co-NP-complete problem even for ( 2 P 5 , K 1 , 5 ) -free graphs. We then show that decision problems Independent Dominating Set , Independent Set , and Dominating Set are NP-complete for the class of well-indumatched graphs. We also show that this class is a co-indumatching hereditary class, i.e., it is closed under deleting the end-vertices of an induced matching along with their neighborhoods, and we characterize well-indumatched graphs in terms of forbidden co-indumatching subgraphs. We prove that recognizing a co-indumatching subgraph is an NP-complete problem. We introduce a perfectly well-indumatched graph, in which every induced subgraph is well-indumatched, and characterize the class of these graphs in terms of forbidden induced subgraphs. Finally, we show that the weighted versions of problems Independent Dominating Set and Independent Set can be solved in polynomial time for perfectly well-indumatched graphs, but problem Dominating Set is NP-complete.

[1]  Elwood S. Buffa,et al.  Graph Theory with Applications , 1977 .

[2]  Gordon F. Royle,et al.  A Characterization of Well-covered Cubic Graphs , 1993 .

[3]  V. Chvátal,et al.  A Note on Well-Covered Graphs , 1993 .

[4]  F. Harary,et al.  On Eulerian and Hamiltonian Graphs and Line Graphs , 1965, Canadian Mathematical Bulletin.

[5]  Ramesh S. Sankaranarayana,et al.  Complexity results for well-covered graphs , 1992, Networks.

[6]  F. Bruce Shepherd,et al.  Bipartite Domination and Simultaneous Matroid Covers , 2003, SIAM J. Discret. Math..

[7]  M. Yannakakis Computing the Minimum Fill-in is NP^Complete , 1981 .

[8]  Andreas Brandstädt,et al.  Dominating Induced Matchings for P7-Free Graphs in Linear Time , 2011, Algorithmica.

[9]  David G. Kirkpatrick,et al.  On the Complexity of General Graph Factor Problems , 1981, SIAM J. Comput..

[10]  Michael Tarsi,et al.  Greedily constructing Hamiltonian paths, Hamiltonian cycles and maximum linear forests , 2007, Discret. Math..

[11]  Gerd Finke,et al.  Approximability results for the maximum and minimum maximal induced matching problems , 2008, Discret. Optim..

[12]  Russell Merris,et al.  Split graphs , 2003, Eur. J. Comb..

[13]  Andreas Brandstädt,et al.  On distance-3 matchings and induced matchings , 2009, Discret. Appl. Math..

[14]  P. Hammer,et al.  Split Graphs Having Dilworth Number Two , 1977, Canadian Journal of Mathematics.

[15]  Peter L. Hammer,et al.  Discrete Applied Mathematics , 1993 .

[16]  Michael D. Plummer,et al.  On well-covered triangulations: Part II , 2009, Discret. Appl. Math..

[17]  Pinar Heggernes,et al.  Minimal triangulations of graphs: A survey , 2006, Discret. Math..

[18]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[19]  Egon Balas,et al.  On graphs with polynomially solvable maximum-weight clique problem , 1989, Networks.

[20]  Игорь Эдмундович Зверович,et al.  Характеризация хорошо укрытых графов в терминах запрещенных костабильных подграфов@@@A characterization of well-covered graphs in terms of forbidden costable subgraphs , 2000 .

[21]  W. T. Tutte Graph Theory , 1984 .

[22]  Claude Berge,et al.  Graphs and Hypergraphs , 2021, Clustering.

[23]  Michael Tarsi,et al.  Well-Covered Claw-Free Graphs , 1996, J. Comb. Theory, Ser. B.

[24]  Moshe Lewenstein,et al.  New results on induced matchings , 2000, Discret. Appl. Math..

[25]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[26]  Jou-Ming Chang,et al.  Induced matchings in asteroidal triple-free graphs , 2003, Discret. Appl. Math..

[27]  Konrad Dabrowski,et al.  New results on maximum induced matchings in bipartite graphs and beyond , 2013, Theor. Comput. Sci..

[28]  I. E. Zverovich A characterization of well-covered graphs in terms of prohibited costable subgraphs , 2000 .

[29]  Michael Tarsi,et al.  Greedily constructing maximal partial f-factors , 2009, Discret. Math..

[30]  Jerzy Topp,et al.  Well Irredundant Graphs , 1995, Discret. Appl. Math..

[31]  M. Plummer Some covering concepts in graphs , 1970 .

[32]  Kathie Cameron,et al.  The graphs with maximum induced matching and maximum matching the same size , 2005, Discret. Math..

[33]  Douglas B. West A short proof of the Berge-Tutte Formula and the Gallai-Edmonds Structure Theorem , 2011, Eur. J. Comb..

[34]  Michael Tarsi,et al.  The Structure of Well-Covered Graphs and the Complexity of Their Recognition Problems , 1997, J. Comb. Theory, Ser. B.

[35]  Vijay V. Vazirani,et al.  NP-Completeness of Some Generalizations of the Maximum Matching Problem , 1982, Inf. Process. Lett..

[36]  Yair Caro,et al.  Local Structure When All Maximal Independent Sets Have Equal Weight , 1998, SIAM J. Discret. Math..

[37]  Masafumi Yamashita,et al.  Modeling K-coteries by well-covered graphs , 1999, Networks.

[38]  Madhav V. Marathe,et al.  The distance-2 matching problem and its relationship to the MAC-Layer capacity of ad hoc wireless networks , 2004, IEEE Journal on Selected Areas in Communications.

[39]  Shuji Tsukiyama,et al.  A New Algorithm for Generating All the Maximal Independent Sets , 1977, SIAM J. Comput..

[40]  Andreas Brandstädt,et al.  The induced matching and chain subgraph cover problems for convex bipartite graphs , 2007, Theor. Comput. Sci..

[41]  Kathie Cameron,et al.  Induced Matchings in Intersection Graphs , 2000, Electron. Notes Discret. Math..

[42]  Yuqin Zhang,et al.  M2-equipackable graphs , 2006, Discret. Appl. Math..

[43]  Richard J. Nowakowski,et al.  A Characterization of Well Covered Graphs of Girth 5 or Greater , 1993, J. Comb. Theory, Ser. B.

[44]  Alan A. Bertossi,et al.  Dominating Sets for Split and Bipartite Graphs , 1984, Inf. Process. Lett..

[45]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[46]  Peter J. Slater,et al.  Fundamentals of domination in graphs , 1998, Pure and applied mathematics.

[47]  Michael D. Plummer,et al.  On well-covered triangulations: Part I , 2003, Discret. Appl. Math..

[48]  Michael D. Plummer,et al.  On well-covered triangulations: Part III , 2010, Discret. Appl. Math..

[49]  Haim Kaplan,et al.  Tractability of Parameterized Completion Problems on Chordal, Strongly Chordal, and Proper Interval Graphs , 1999, SIAM J. Comput..

[50]  Dieter Rautenbach,et al.  Some results on graphs without long induced paths , 2003, Inf. Process. Lett..

[51]  Kathie Cameron,et al.  Finding a maximum induced matching in weakly chordal graphs , 2003, Discret. Math..

[52]  Angelika Steger,et al.  On induced matchings , 1993, Discret. Math..

[53]  Yair Caro,et al.  Recognizing Greedy Structures , 1996, J. Algorithms.

[54]  Andreas Brandstädt,et al.  Maximum Induced Matchings for Chordal Graphs in Linear Time , 2008, Algorithmica.

[55]  Udi Rotics,et al.  Finding Maximum Induced Matchings in Subclasses of Claw-Free and P5-Free Graphs, and in Graphs with Matching and Induced Matching of Equal Maximum Size , 2003, Algorithmica.