A structure theorem for graphs with no cycle with a unique chord and its consequences

We give a structural description of the class C of graphs that do not contain a cycle with a unique chord as an induced subgraph. Our main theorem states that any connected graph in C is a either in some simple basic class or has a decomposition. Basic classes are cliques, bipartite graphs with one side containing only nodes of degree two and induced subgraph of the famous Heawood or Petersen graph. Decompositions are node cutsets consisting of one or two nodes and edge cutsets called 1-joins. Our decomposition theorem actually gives a complete structure theorem for C, i.e. every graph in C can be built from basic graphs that can be explicitly constructed, and gluing them together by prescribed composition operations ; and all graphs built this way are in C. This has several consequences : an O(nm)-time algorithm to decide whether a graph is in C, an O(n+m)-time algorithm that finds a maximum clique of any graph in C and an O(nm)-time coloring algorithm for graphs in C. We prove that every graph in C is either 3-colorable or has a coloring with ? colors where ? is the size of a largest clique. The problem of finding a maximum stable set for a graph in C is known to be NP-hard.

[1]  Klaus Truemper,et al.  Alpha-balanced graphs and matrices and GF(3)-representability of matroids , 1982, J. Comb. Theory, Ser. B.

[2]  Gérard Cornuéjols,et al.  Decomposing Berge Graphs Containing Proper Wheels , 2002 .

[3]  Mihalis Yannakakis.  On a class of totally unimodular matrices , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[4]  P. Erdös,et al.  Graph Theory and Probability , 1959 .

[5]  Vasek Chvátal,et al.  Star-cutsets and perfect graphs , 1985, J. Comb. Theory, Ser. B.

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

[7]  P. Seymour,et al.  Excluding induced subgraphs , 2006 .

[8]  Robert E. Tarjan,et al.  Depth-First Search and Linear Graph Algorithms , 1972, SIAM J. Comput..

[9]  Nicolas Trotignon,et al.  Detecting induced subgraphs , 2007, Discret. Appl. Math..

[10]  Daniel Bienstock,et al.  On the complexity of testing for odd holes and induced odd paths , 1991, Discret. Math..

[11]  Robert E. Tarjan,et al.  Dividing a Graph into Triconnected Components , 1973, SIAM J. Comput..

[12]  Paul D. Seymour,et al.  Even pairs in Berge graphs , 2009, J. Comb. Theory, Ser. B.

[13]  W. Cunningham.  Decomposition of Directed Graphs , 1982 .

[14]  Gérard Cornuéjols,et al.  Even and odd holes in cap-free graphs , 1999 .

[15]  Ingo Schiermeyer,et al.  Vertex Colouring and Forbidden Subgraphs – A Survey , 2004, Graphs Comb..

[16]  Gérard Cornuéjols,et al.  Balanced matrices , 2006, Discret. Math..

[17]  Ryan B. Hayward,et al.  Weakly triangulated graphs , 1985, J. Comb. Theory B.

[18]  Liping Sun.  Two classes of perfect graphs , 1991, J. Comb. Theory, Ser. B.

[19]  Ryan B. Hayward.  Discs in unbreakable graphs , 1995, Graphs Comb..

[20]  Michele Conforti,et al.  Structural properties and recognition of restricted and strongly unimodular matrices , 1987, Math. Program..

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

[22]  P. J. Heawood.  Map-Colour Theorem , 1949 .

[23]  Nicolas Trotignon,et al.  Algorithms for Square-3PC(., .)-Free Berge Graphs , 2008, SIAM J. Discret. Math..

[24]  Chính T. Hoàng,et al.  Optimizing weakly triangulated graphs , 1990, Graphs Comb..

[25]  A. Gyárfás.  Problems from the world surrounding perfect graphs , 1987 .

[26]  Paul D. Seymour,et al.  The three-in-a-tree problem , 2010, Comb..

[27]  Gérard Cornuéjols,et al.  Even and odd holes in cap-free graphs , 1999, J. Graph Theory.

[28]  Paul D. Seymour,et al.  Recognizing Berge Graphs , 2005, Comb..

[29]  Henry Meyniel,et al.  A New Property of Critical Imperfect Graphs and some Consequences , 1987, Eur. J. Comb..

[30]  Florian Roussel,et al.  About Skew Partitions in Minimal Imperfect Graphs , 2001, J. Comb. Theory, Ser. B.

[31]  Myriam Preissmann,et al.  On the NP-completeness of the k-colorability problem for triangle-free graphs , 1996, Discret. Math..

[32]  Elias Dahlhaus,et al.  Parallel Algorithms for Hierarchical Clustering and Applications to Split Decomposition and Parity Graph Recognition , 2000, J. Algorithms.

[33]  Nicolas Trotignon,et al.  A class of perfectly contractile graphs , 2006, J. Comb. Theory, Ser. B.

[34]  Nicolas Trotignon,et al.  Algorithms for Perfectly Contractile Graphs , 2005, SIAM J. Discret. Math..