Well-Quasi-Ordering versus Clique-Width: New Results on Bigenic Classes

Daligault, Rao and Thomasse conjectured that if a hereditary class of graphs is well-quasi-ordered by the induced subgraph relation then it has bounded clique-width. Lozin, Razgon and Zamaraev recently showed that this conjecture is not true for infinitely defined classes. For finitely defined classes the conjecture is still open. It is known to hold for classes of graphs defined by a single forbidden induced subgraph H, as such graphs are well-quasi-ordered and are of bounded clique-width if and only if H is an induced subgraph of \(P_4\). For bigenic classes of graphs i.e. ones defined by two forbidden induced subgraphs there are several open cases in both classifications. We reduce the number of open cases for well-quasi-orderability of such classes from 12 to 9. Our results agree with the conjecture and imply that there are only two remaining cases to verify for bigenic classes.

[1]  Konrad Dabrowski,et al.  Bounding Clique-Width via Perfect Graphs , 2014, LATA.

[2]  Bruno Courcelle,et al.  Linear Time Solvable Optimization Problems on Graphs of Bounded Clique-Width , 2000, Theory of Computing Systems.

[3]  Andreas Brandstädt,et al.  Gem- And Co-Gem-Free Graphs Have Bounded Clique-Width , 2004, Int. J. Found. Comput. Sci..

[4]  Philippe Schnoebelen,et al.  Well-structured transition systems everywhere! , 2001, Theor. Comput. Sci..

[5]  Graham Higman,et al.  Ordering by Divisibility in Abstract Algebras , 1952 .

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

[7]  Bruno Courcelle.  Clique-width and edge contraction , 2014, Inf. Process. Lett..

[8]  Bruno Courcelle,et al.  Upper bounds to the clique width of graphs , 2000, Discret. Appl. Math..

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

[10]  Konrad Dabrowski,et al.  Clique-Width of Graph Classes Defined by Two Forbidden Induced Subgraphs , 2014, Comput. J..

[11]  B. Mohar,et al.  Graph Minors , 2009 .

[12]  Peter Damaschke,et al.  Induced subgraphs and well-quasi-ordering , 1990, J. Graph Theory.

[13]  Hans L. Bodlaender.  A linear time algorithm for finding tree-decompositions of small treewidth , 1993, STOC '93.

[14]  Daniël Paulusma,et al.  Well-Quasi-Ordering versus Clique-Width: New Results on Bigenic Classes , 2018 .

[15]  Egon Wanke,et al.  How to Solve NP-hard Graph Problems on Clique-Width Bounded Graphs in Polynomial Time , 2001, WG.

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

[17]  Joseph B. Kruskal,et al.  The Theory of Well-Quasi-Ordering: A Frequently Discovered Concept , 1972, J. Comb. Theory A.

[18]  Paul D. Seymour,et al.  Graph Minors. XX. Wagner's conjecture , 2004, J. Comb. Theory B.

[19]  Vadim V. Lozin,et al.  Labelled Induced Subgraphs and Well-Quasi-Ordering , 2015, Order.

[20]  Udi Rotics,et al.  Edge dominating set and colorings on graphs with fixed clique-width , 2003, Discret. Appl. Math..

[21]  Stéphan Thomassé,et al.  Well-Quasi-Order of Relabel Functions , 2010, Order.

[22]  Daniël Paulusma,et al.  Colouring diamond-free graphs , 2015, J. Comput. Syst. Sci..

[23]  Michaël Rao,et al.  MSOL partitioning problems on graphs of bounded treewidth and clique-width , 2007, Theor. Comput. Sci..

[24]  Vadim V. Lozin,et al.  Well-quasi-ordering Does Not Imply Bounded Clique-width , 2015, WG.

[25]  Konrad Dabrowski,et al.  Clique-Width of Graph Classes Defined by Two Forbidden Induced Subgraphs , 2015, CIAC.

[26]  Paul D. Seymour,et al.  Graph minors. IV. Tree-width and well-quasi-ordering , 1990, J. Comb. Theory, Ser. B.

[27]  Bruno Courcelle,et al.  The monadic second-order logic of graphs III: tree-decompositions, minor and complexity issues , 1992, RAIRO Theor. Informatics Appl..

[28]  Vadim V. Lozin,et al.  Two forbidden induced subgraphs and well-quasi-ordering , 2011, Discret. Math..