Even‐hole‐free graphs part I: Decomposition theorem

We prove a decomposition theorem for even-hole-free graphs. The decompositions used are 2-joins and star, double-star and triple-star cutsets. This theorem is used in the second part of this paper to obtain a polytime recognition algorithm for even-hole-free graphs.

[1]  Jim Geelen,et al.  Matchings, Matroids and Unimodular Matrices , 1995 .

[2]  Michele Conforti,et al.  A Theorem of Truemper , 1998, IPCO.

[3]  Gérard Cornuéjols,et al.  Compositions for perfect graphs , 1985, Discret. Math..

[4]  Oscar Porto.  Even Induced Cycles in Planar Graphs , 1992, LATIN.

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

[6]  Gérard Cornuéjols,et al.  Even‐hole‐free graphs part II: Recognition algorithm , 2002, J. Graph Theory.

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

[8]  Gérard Cornuéjols,et al.  Decomposition of Balanced Matrices , 1999, J. Comb. Theory, Ser. B.

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

[10]  M. Burlet,et al.  Polynomial algorithm to recognize a Meyniel graph , 1984 .

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

[12]  Kathryn Fraughnaugh,et al.  Introduction to graph theory , 1973, Mathematical Gazette.

[13]  Gérard Cornuéjols,et al.  Triangle-free graphs that are signable without even holes , 2000 .

[14]  Bruce A. Reed,et al.  beta-Perfect Graphs , 1996, J. Comb. Theory, Ser. B.

[15]  R. Bixby.  A Composition for Perfect Graphs , 1984 .

[16]  Gérard Cornuéjols,et al.  Balanced 0-1 Matrices I. Decomposition , 2001, J. Comb. Theory, Ser. B.

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

[18]  S. E. Markosyan,et al.  ω-Perfect graphs , 1990 .