On the structure of (P5, gem)-free graphs

We give a complete structure description of (P"5,gem)-free graphs. By the results of a related paper, this implies bounded clique width for this graph class. Hereby, as usual, the P"5 is the induced path with five vertices a,b,c,d,e and four edges ab,bc,cd,de, and the gem consists of a P"4a,b,c,d with edges ab,bc,cd plus a universal vertex e adjacent to a,b,c,d.

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

[2]  M. Golumbic.  Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57) , 2004 .

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

[4]  C. Lekkeikerker,et al.  Representation of a finite graph by a set of intervals on the real line , 1962 .

[5]  Robert E. Tarjan,et al.  Algorithmic Aspects of Vertex Elimination on Graphs , 1976, SIAM J. Comput..

[6]  Bruno Courcelle,et al.  Handle-Rewriting Hypergraph Grammars , 1993, J. Comput. Syst. Sci..

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

[8]  Derek G. Corneil,et al.  Complement reducible graphs , 1981, Discret. Appl. Math..

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

[10]  Robert E. Tarjan,et al.  Simple Linear-Time Algorithms to Test Chordality of Graphs, Test Acyclicity of Hypergraphs, and Selectively Reduce Acyclic Hypergraphs , 1984, SIAM J. Comput..

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

[12]  M. Yannakakis.  The Complexity of the Partial Order Dimension Problem , 1982 .

[13]  Sophie Tison.  Trees in Algebra and Programming — CAAP'94 , 1994, Lecture Notes in Computer Science.

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

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

[16]  Bruce A. Reed,et al.  Some classes of perfectly orderable graphs , 1989, J. Graph Theory.

[17]  Jens Gustedt,et al.  Efficient and Practical Algorithms for Sequential Modular Decomposition , 2001, J. Algorithms.

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