Independent domination in finitely defined classes of graphs: Polynomial algorithms

We study the problem of finding in a graph an inclusionwise maximal independent set of minimum cardinality, known as minimum maximal independent set or independent dominating set problem. This is one of the hardest problems in algorithmic graph theory. In particular, restricted to the class of so called SAT-graphs, this problem coincides with satisfiability, the central problem of theoretical computer science. The class of SAT-graphs, and many other important graph classes, such as graphs of bounded vertex degree or line graphs, can be characterized by finitely many forbidden induced subgraphs. We call such classes finitely defined. The paper [R. Boliac and V. Lozin, Independent domination in finitely defined classes of graphs, Theoretical Computer Science, 301 (2003) 271-284] reveals various conditions under which the independent dominating set problem remains NP-hard in a finitely defined class. In the present paper, we identify a number of finitely defined classes where the problem admits polynomial-time solutions. In particular, we prove that the problem is solvable in polynomial time in the class of P 2 + P 3 -free graphs by correcting a mistake of the first two authors of the paper in their earlier solution. This result is in a sharp contrast with the fact that in the class of P 3 + P 3 -free graphs the problem is known to be NP-hard, since this class contains all SAT-graphs.

[1]  Andrzej Ehrenfeucht,et al.  An O(n²) Divide-and-Conquer Algorithm for the Prime Tree Decomposition of Two-Structures and Modular Decomposition of Graphs , 1994, J. Algorithms.

[2]  C. De Simone,et al.  On the vertex packing problem , 1993, Graphs Comb..

[3]  Peter Damaschke,et al.  Domination in Convex and Chordal Bipartite Graphs , 1990, Inf. Process. Lett..

[4]  Vadim V. Lozin,et al.  On Independent Vertex Sets in Subclasses of Apple-Free Graphs , 2010, Algorithmica.

[5]  T. Gallai.  Transitiv orientierbare Graphen , 1967 .

[6]  Bruno Courcelle,et al.  Linear Time Solvable Optimization Problems on Graphs of Bounded Clique Width , 1998, WG.

[7]  Ingo Schiermeyer,et al.  On maximum independent sets in P5-free graphs , 2010, Discret. Appl. Math..

[8]  Vadim V. Lozin,et al.  Maximum independent sets in subclasses of P5-free graphs , 2009, Inf. Process. Lett..

[9]  Alain Hertz,et al.  P5-free augmenting graphs and the maximum stable set problem , 2002, Discret. Appl. Math..

[10]  Martin Farber,et al.  On diameters and radii of bridged graphs , 1989, Discret. Math..

[11]  Vadim V. Lozin,et al.  An augmenting graph approach to the stable set problem in P5-free graphs , 2003, Discret. Appl. Math..

[12]  Raffaele Mosca.  Some results on maximum stable sets in certain P5-free graphs , 2003, Discret. Appl. Math..

[13]  Vadim V. Lozin,et al.  Augmenting graphs for independent sets , 2004, Discret. Appl. Math..

[14]  Frank Harary,et al.  Graph Theory , 2016 .

[15]  Gerard J. Chang,et al.  The weighted independent domination problem is NP-complete for chordal graphs , 2004, Discret. Appl. Math..

[16]  Raffaele Mosca.  Some observations on maximum weight stable sets in certain P , 2008, Eur. J. Oper. Res..

[17]  Vadim V. Lozin,et al.  Independent sets in extensions of 2K2-free graphs , 2005, Discret. Appl. Math..

[18]  Vadim V. Lozin,et al.  Independent domination in finitely defined classes of graphs , 2003, Theor. Comput. Sci..

[19]  Igor E. Zverovich,et al.  Satgraphs and independent domination. Part 1 , 2006, Theor. Comput. Sci..

[20]  Martin Farber,et al.  Independent domination in chordal graphs , 1982, Oper. Res. Lett..

[21]  Jeremy P. Spinrad,et al.  Modular decomposition and transitive orientation , 1999, Discret. Math..

[22]  Vadim V. Lozin,et al.  On finding augmenting graphs , 2008, Discret. Appl. Math..

[23]  Frédéric Maffray,et al.  Stable sets in k-colorable P5-free graphs , 2009, Inf. Process. Lett..

[24]  Jeremy P. Spinrad,et al.  On algorithms for (P5, gem)-free graphs , 2005, Theor. Comput. Sci..

[25]  Michel Habib,et al.  On the X-join decomposition for undirected graphs , 1979, Discret. Appl. Math..

[26]  Mihalis Yannakakis,et al.  Edge Dominating Sets in Graphs , 1980 .

[27]  R. Möhring.  Algorithmic Aspects of Comparability Graphs and Interval Graphs , 1985 .