On Independent Vertex Sets in Subclasses of Apple-Free Graphs

In a finite undirected graph, an apple consists of a chordless cycle of length at least 4, and an additional vertex which is not in the cycle and sees exactly one of the cycle vertices. A graph is apple-free if it contains no induced subgraph isomorphic to an apple. Apple-free graphs are a common generalization of chordal graphs, claw-free graphs and cographs and occur in various papers. The Maximum Weight Independent Set (MWS) problem is efficiently solvable on chordal graphs, on cographs as well as on claw-free graphs. In this paper, we obtain partial results on some subclasses of apple-free graphs where our results show that the MWS problem is solvable in polynomial time. The main tool is a combination of clique separators with modular decomposition.Our algorithms are robust in the sense that there is no need to recognize whether the input graph is in the given graph class; the algorithm either solves the MWS problem correctly or detects that the input graph is not in the given class.

[1]  F. Radermacher,et al.  Substitution Decomposition for Discrete Structures and Connections with Combinatorial Optimization , 1984 .

[2]  D. R. Lick,et al.  The Theory and Applications of Graphs. , 1983 .

[3]  Feodor F. Dragan,et al.  Linear Time Algorithms for Hamiltonian Problems on (Claw, Net)-Free Graphs , 1999, SIAM J. Comput..

[4]  A. Brandstädt,et al.  Graph Classes: A Survey , 1987 .

[5]  Martin Milanič.  Algorithmic developments and complexity results for finding maximum and exact independent sets in graphs , 2007 .

[6]  Jeremy P. Spinrad,et al.  Efficient graph representations , 2003, Fields Institute monographs.

[7]  Raffaele Mosca,et al.  Stable sets of maximum weight in (P7, banner)-free graphs , 2008, Discret. Math..

[8]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..

[9]  Vadim V. Lozin,et al.  A note on alpha-redundant vertices in graphs , 2001, Discret. Appl. Math..

[10]  S. Whitesides.  A Method for Solving Certain Graph Recognition and Optimization Problems, with Applications to Perfect Graphs , 1982 .

[11]  S. Poljak.  A note on stable sets and colorings of graphs , 1974 .

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

[13]  Peter L. Hammer,et al.  The struction of a graph: Application toCN-free graphs , 1985, Comb..

[14]  Vadim V. Lozin,et al.  Maximum independent sets in graphs of low degree , 2007, SODA '07.

[15]  Vadim V. Lozin,et al.  Robust Algorithms for the Stable Set Problem , 2003, Graphs Comb..

[16]  Stephan Olariu,et al.  The Strong Perfect Graph Conjecture for pan-free graphs , 1989, J. Comb. Theory, Ser. B.

[17]  Vadim V. Lozin,et al.  Stability in P5- and banner-free graphs , 2000, Eur. J. Oper. Res..

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

[19]  V. Chvátal,et al.  Topics on perfect graphs , 1984 .

[20]  A. Tamura,et al.  A revision of Minty's algorithm for finding a maximum weight stable set of a claw-free graph , 2001 .

[21]  M. Golumbic.  Algorithmic graph theory and perfect graphs , 1980 .

[22]  Paul D. Seymour,et al.  The structure of claw-free graphs , 2005, BCC.

[23]  Raffaele Mosca.  Stable Sets in Certain P6-free Graphs , 1999, Discret. Appl. Math..

[24]  Andreas Brandstädt,et al.  On clique separators, nearly chordal graphs, and the Maximum Weight Stable Set Problem , 2007, Theor. Comput. Sci..

[25]  Andreas Brandstädt,et al.  New applications of clique separator decomposition for the Maximum Weight Stable Set problem , 2007, Theor. Comput. Sci..

[26]  Vadim V. Lozin,et al.  Stable sets in two subclasses of banner-free graphs , 2003, Discret. Appl. Math..

[27]  Robert E. Tarjan,et al.  Decomposition by clique separators , 1985, Discret. Math..

[28]  Feodor F. Dragan,et al.  On linear and circular structure of (claw, net)-free graphs , 2003, Discret. Appl. Math..

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

[30]  P. Seymour,et al.  The Strong Perfect Graph Theorem , 2002, math/0212070.

[31]  Lorna Stewart,et al.  A Linear Recognition Algorithm for Cographs , 1985, SIAM J. Comput..

[32]  George J. Minty,et al.  On maximal independent sets of vertices in claw-free graphs , 1980, J. Comb. Theory, Ser. B.

[33]  B. S Webb,et al.  Surveys in combinatorics 2005 , 2005 .

[34]  Alexander K. Kelmans.  On Hamiltonicity of {claw, net}-free graphs , 2006, Discret. Math..