New applications of clique separator decomposition for the Maximum Weight Stable Set problem

Graph decompositions such as decomposition by clique separators and modular decomposition are of crucial importance for designing efficient graph algorithms. Clique separators in graphs were used by Tarjan as a divide-and-conquer approach for solving various problems such as the Maximum Weight Stable Set (MWS) Problem, Coloring and Minimum Fill-in. The basic tool is a decomposition tree of the graph whose leaves have no clique separator (so-called atoms), and the problem can be solved efficiently on the graph if it is efficiently solvable on its atoms. We give new examples where the clique separator decomposition works well for the MWS problem which also improves and extends various recently published results. In particular, we describe the atom structure for some new classes of graphs whose atoms are P5-free (the P5 is the induced path with 5 vertices) and obtain new polynomial time results for MWS.

[1]  Raffaele Mosca,et al.  On (P5, diamond)-free graphs , 2002, Discret. Math..

[2]  Caterina De Simone,et al.  Stability Number of Bull- and Chair-Free Graphs , 1993, Discret. Appl. Math..

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

[4]  Pinar Heggernes,et al.  Recognizing Weakly Triangulated Graphs by Edge Separability , 2000, Nord. J. Comput..

[5]  Dennis Saleh.  Zs , 2001 .

[6]  Stephen H. Unger,et al.  Minimizing the Number of States in Incompletely Specified Sequential Switching Functions , 1959, IRE Trans. Electron. Comput..

[7]  Norishige Chiba,et al.  Arboricity and Subgraph Listing Algorithms , 1985, SIAM J. Comput..

[8]  Vadim V. Lozin,et al.  On the stable set problem in special P5-free graphs , 2003, Discret. Appl. Math..

[9]  Peter L. Hammer,et al.  Stability in CAN-free graphs , 1985, J. Comb. Theory, Ser. B.

[10]  Matthias MIDDENDORF,et al.  On the complexity of recognizing perfectly orderable graphs , 1990, Discret. Math..

[11]  Panos M. Pardalos,et al.  The maximum clique problem , 1994, J. Glob. Optim..

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

[13]  Andreas Brandst.  (P 5 ,diamond)-free graphs revisited: structure and linear time optimization , 2004 .

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

[15]  V. E. Alekseev,et al.  Polynomial algorithm for finding the largest independent sets in graphs without forks , 2004, Discret. Appl. Math..

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

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

[18]  Andreas Brandst.  On clique separators, nearly chordal graphs, and the Maximum Weight Stable Set Problem , 2007 .

[19]  Najiba Sbihi,et al.  Algorithme de recherche d'un stable de cardinalite maximum dans un graphe sans etoile , 1980, Discret. Math..

[20]  Andreas Brandstädt,et al.  On alpha-redundant vertices in P5-free graphs , 2002, Inf. Process. Lett..

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

[22]  Owen J. Murphy,et al.  Computing independent sets in graphs with large girth , 1992, Discret. Appl. Math..

[23]  Chính T. Hoàng,et al.  Efficient Algorithms for Minimum Weighted Colouring of Some Classes of Perfect Graphs , 1994, Discret. Appl. Math..

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

[25]  Andreas Brandstädt,et al.  On the structure and stability number of P5- and co-chair-free graphs , 2003, Discret. Appl. Math..

[26]  Peter L. Hammer,et al.  On the Stability Number of Claw-free P5-free and More General Graphs , 1999, Discret. Appl. Math..

[27]  Vassilis Giakoumakis,et al.  Weighted Parameters in (P5, P5)-free Graphs , 1997, Discret. Appl. Math..

[28]  Dieter Kratsch,et al.  On the structure of (P5, gem)-free graphs , 2005, Discret. Appl. Math..

[29]  Andreas Brandstädt,et al.  Structure and stability number of chair-, co-P- and gem-free graphs revisited , 2003, Inf. Process. Lett..

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

[31]  Peter L. Hammer,et al.  Stability in Circular Arc Graphs , 1988, J. Algorithms.

[32]  Dominique de Werra,et al.  Four classes of perfectly orderable graphs , 1987, J. Graph Theory.

[33]  Andreas Brandstädt,et al.  On clique separators, nearly chordal graphs, and the Maximum Weight Stable Set Problem , 2005, Theor. Comput. Sci..

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

[35]  Jeremy P. Spinrad,et al.  Weakly chordal graph algorithms via handles , 2000, SODA '00.

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

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

[38]  Alain Hertz,et al.  A Transformation Which Preserves the Clique Number , 2001, J. Comb. Theory, Ser. B.

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

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

[41]  V. E. Alekseev,et al.  On easy and hard hereditary classes of graphs with respect to the independent set problem , 2003, Discret. Appl. Math..

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

[43]  Andreas Brandstädt,et al.  Efficient robust algorithms for the Maximum Weight Stable Set Problem in chair-free graph classes , 2004, Inf. Process. Lett..

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

[45]  Andreas Brandstädt,et al.  Chordal co-gem-free and (P5, gem)-free graphs have bounded clique-width , 2005, Discret. Appl. Math..

[46]  Raffaele Mosca,et al.  Polynomial Algorithms for the Maximum Stable Set Problem on Particular Classes of P_5-Free Graphs , 1997, Inf. Process. Lett..

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

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

[49]  Andreas Brandstädt,et al.  Stability number of bull- and chair-free graphs revisited , 2003, Discret. Appl. Math..

[50]  Andreas Brandstädt,et al.  Maximum Weight Stable Set on graphs without claw and co-claw (and similar graph classes) can be solved in linear time , 2002, Inf. Process. Lett..

[51]  Zsolt Tuza,et al.  An upper bound on the number of cliques in a graph , 1993, Networks.

[52]  Gary MacGillivray,et al.  Domination in Convex Bipartite and Convex-round Graphs , 1999 .

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

[54]  Mihalis Yannakakis,et al.  On Generating All Maximal Independent Sets , 1988, Inf. Process. Lett..

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

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

[57]  Jeremy P. Spinrad,et al.  Algorithms for Weakly Triangulated Graphs , 1995, Discret. Appl. Math..

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

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

[60]  Derek G. Corneil.  The complexity of generalized clique packing , 1985, Discret. Appl. Math..

[61]  Florian Roussel,et al.  Holes and Dominoes in Meyniel Graphs , 1999, Int. J. Found. Comput. Sci..

[62]  Jeremy P. Spinrad,et al.  Linear Time Algorithms for Some NP-Complete Problems on (P5, Gem)-Free Graphs , 2003, FCT.

[63]  Andreas Brandstädt,et al.  (P5, diamond)-free graphs revisited: structure and linear time optimization , 2004, Discret. Appl. Math..

[64]  Caterina De Simone,et al.  A Nice Class for the Vertex Packing Problem , 1997, Discret. Appl. Math..