On rank-width of even-hole-free graphs

We present a class of (diamond, even hole)-free graphs with no clique cutset that has unbounded rank-width. In general, even-hole-free graphs have unbounded rank-width, because chordal graphs are even-hole-free. A.A. da Silva, A. Silva and C. Linhares-Sales (2010) showed that planar even-hole-free graphs have bounded rank-width, and N.K. Le (2016) showed that even-hole-free graphs with no star cutset have bounded rank-width. A natural question is to ask, whether even-hole-free graphs with no clique cutsets have bounded rank-width. Our result gives a negative answer. Hence we cannot apply Courcelle and Makowsky's meta-theorem which would provide efficient algorithms for a large number of problems, including the maximum independent set problem, whose complexity remains open for (diamond, even hole)-free graphs.

[1]  Kristina Vuskovic,et al.  The world of hereditary graph classes viewed through Truemper configurations , 2013, Surveys in Combinatorics.

[2]  Bert Gerards,et al.  Stable sets and graphs with no even holes , 2013, Math. Program..

[3]  Kristina Vuskovic,et al.  Even-hole-free graphs that do not contain diamonds: A structure theorem and its consequences , 2009, J. Comb. Theory, Ser. B.

[4]  Udi Rotics,et al.  On the Clique-Width of Some Perfect Graph Classes , 2000, Int. J. Found. Comput. Sci..

[5]  Kristina Vušković,et al.  EVEN-HOLE-FREE GRAPHS: A SURVEY , 2010 .

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

[7]  Nicolas Trotignon,et al.  Vertex elimination orderings for hereditary graph classes , 2012, Discret. Math..

[8]  Cláudia Linhares Sales,et al.  A bound on the treewidth of planar even-hole-free graphs , 2010, Discret. Appl. Math..

[9]  Bruno Courcelle,et al.  Graph Rewriting: An Algebraic and Logic Approach , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[10]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs. I. Recognizable Sets of Finite Graphs , 1990, Inf. Comput..

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

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

[13]  Paul D. Seymour,et al.  Graph Minors. II. Algorithmic Aspects of Tree-Width , 1986, J. Algorithms.

[14]  Paul D. Seymour,et al.  Approximating clique-width and branch-width , 2006, J. Comb. Theory B.

[15]  Shenwei Huang,et al.  Structure and algorithms for (cap, even hole)-free graphs , 2016, Discret. Math..

[16]  Kristina Vuskovic,et al.  Triangulated neighborhoods in even-hole-free graphs , 2007, Discret. Math..

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

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

[19]  Da Silva,et al.  Even-hole-free graphs , 2008 .

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

[21]  Bruce A. Reed,et al.  Bisimplicial vertices in even-hole-free graphs , 2008, J. Comb. Theory, Ser. B.

[22]  T. Kloks.  Graphs without even holes or diamonds , 2011, 1104.3920.

[23]  Kristina Vuskovic,et al.  Decomposition of even-hole-free graphs with star cutsets and 2-joins , 2013, J. Comb. Theory, Ser. B.