Characterizations for co-graphs defined by restricted NLC-width or clique-width operations

In this paper we characterize subclasses of co-graphs defined by restricted NLC-width operations and subclasses of co-graphs defined by restricted clique-width operations. We show that a graph has NLCT-width 1 if and only if it is (C"4,P"4)-free. Since (C"4,P"4)-free graphs are exactly trivially perfect graphs, the set of graphs of NLCT-width 1 is equal to the set of trivially perfect graphs, and a recursive definition for trivially perfect graphs follows. Further we show that a graph has linear NLC-width 1 if and only if is (C"4,P"4,2K"2)-free. This implies that the set of graphs of linear NLC-width 1 is equal to the set of threshold graphs. We also give forbidden induced subgraph characterizations for co-graphs defined by restricted clique-width operations using P"4, 2K"2, and co-2P"3.

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

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

[3]  Martin Charles Golumbic,et al.  Trivially perfect graphs , 1978, Discret. Math..

[4]  Egon Wanke,et al.  Vertex disjoint paths on clique-width bounded graphs , 2004, Theor. Comput. Sci..

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

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

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

[8]  Egon Wanke,et al.  On the relationship between NLC-width and linear NLC-width , 2005, Theor. Comput. Sci..

[9]  Egon Wanke,et al.  Minimizing NLC-Width is NP-Complete , 2005, WG.

[10]  Öjvind Johansson.  NLC2-Decomposition in Polynomial Time , 1999, WG.

[11]  E. S. Wolk.  The comparability graph of a tree , 1962 .

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

[13]  Egon Wanke,et al.  k-NLC Graphs and Polynomial Algorithms , 1994, Discret. Appl. Math..

[14]  N. Mahadev,et al.  Threshold graphs and related topics , 1995 .

[15]  P. Hammer,et al.  Aggregation of inequalities in integer programming. , 1975 .

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

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