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Abstract
We define rank-width of graphs to investigate clique-width. Rank-width is a complexity measure of decomposing a graph in a kind of tree-structure, called a rank-decomposition. We show that graphs have bounded rank-width if and only if they have bounded clique-width.
It is unknown how to recognize graphs of clique-width at most k for fixed k > 3 in polynomial time. However, we find an algorithm recognizing graphs of rank-width at most k, by combining following three ingredients.
First, we construct a polynomial-time algorithm, for fixed k , that confirms rank-width is larger than k or outputs a rank-decomposition of width at most f (k) for some function f. It was known that many hard graph problems have polynomial-time algorithms for graphs of bounded clique-width, however, requiring a given decomposition corresponding to clique-width (k-expression ); we remove this requirement.
Second, we define graph vertex-minors which generalizes matroid minors, and prove that if {G1, G2,…} is an infinite sequence of graphs of bounded rank-width, then there exist i < j such that Gi is isomorphic to a vertex-minor of Gj. Consequently there is a finite list [special characters omitted] of graphs such that a graph has rank-width at most k if and only if none of its vertex-minors are isomorphic to a graph in [special characters omitted].
Finally we construct, for fixed graph H, a modulo-2 counting monadic second-order logic formula expressing a graph contains a vertex-minor isomorphic to H. It is known that such logic formulas are solvable in linear time on graphs of bounded clique-width if the k-expression is given as an input.
Another open problem in the area of clique-width is Seese's conjecture; if a set of graphs have an algorithm to answer whether a given monadic second-order logic formula is true for all graphs in the set, then it has bounded rank-width. We prove a weaker statement; if the algorithm answers for all modulo-2 counting monadic second-order logic formulas, then the set has bounded rank-width.
