Polynomial time recognition of clique-width ≤ 3 graphs

Abstract

The Clique-width of a graph is an invariant which measures the complexity of the graph structures. A graph of bounded tree-width is also of bounded Clique-width (but not the converse). For graphs G of bounded Clique-width, given the bounded width decomposition of G, every optimization, enumeration or evaluation problem that can be defined by a Monadic Second Order Logic formula using quantifiers on vertices but not on edges, can be solved in polynomial time. This is reminiscent of the situation for graphs of bounded tree-width, where the same statement holds even if quantifiers are also allowed on edges. Thus, graphs of bounded Clique-width are a larger class than graphs of bounded tree-width, on which we can resolve fewer, but still many, optimization problems efficiently. In this paper we present the first polynomial time algorithm (O(n2m)) to recognize graphs of Clique-width at most 3. © Springer-Verlag Berlin Heidelberg 2000.