Graph Decomposition Using Node Labels

Dynamic programming is a method in which the solution to a computational problem is found by combination of already obtained solutions to subproblems. This method can be applied to problems on graphs (nodes connected by edges). The graph of interest must then be broken down into successively smaller parts according to some suitable principle. The thesis studies two graph algebras introduced for this purpose (Wanke 1994; Courcelle and Olariu 1994). Nodes have labels, and roughly, the operations are union, edge drawing, and relabeling. Once two nodes have acquired the same label, they can no longer be differentiated at subsequent edge drawings and relabelings. NLC-decompositions and clique-decompositions are derivations of graphs from their individual nodes using the operations of the respective algebra. The width of such a decomposition is the number of distinct labels used, and the minimum width among all decompositions of a particular graph is known as its NLC-width and clique-width respectively. Numerous NP-hard graph problems can be solved efficiently with dynamic programming if a decomposition of small width is given for the input graph. The two algebras generalize the class of cographs, and can refine the modular decomposition of a graph. Moreover, they are related to graph grammars, and the above decompositions of a graph can be seen as possible growth histories for it under certain inheritance principles. The thesis gives a simple description of this. It then shows: • A clique-decomposition can be transformed into an equivalent NLCdecomposition of the same width, and an NLC-decomposition of width k can be transformed into an equivalent clique-decomposition of width at most 2k. • A randomly generated unlabeled graph on a set of n nodes is very likely to have NLC-width and clique-width almost n/2 and n respectively, provided that n is sufficiently large. • If a graph has NLC-width 2, a corresponding decomposition can be found in O(n log n) time. • An NLC-decomposition of width at most log n times the optimal width k can be found in O(n) time.