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Abstract: The d-dimensional hypercube, , is the graph on vertices, which correspond to the d-vectors whose components are either 0 or 1, two of the vertices being adjacent when they differ in just one coordinate. The notion of Hamming graphs (denoted by ) generalizes the notion of hypercubes: The vertices correspond to the d-vectors where the components are from the set , and two of the vertices are adjacent if and only if the corresponding vectors differ in exactly one component. In this paper we show that the and the . [Copyright &y& Elsevier]

Abstract: Motivated by the desire to speed up dynamic programming algorithms for graphs of bounded treewidth, we initiate a study of the tradeoff between width and pathwidth of tree-decompositions. We therefore investigate the catwidth parameter which is the minimum width of any tree-decomposition of a graph G when the pathwidth of the tree T is 1. The catwidth parameter lies between the treewidth and the pathwidth of the graph, , and just as treewidth relates to chordal graphs and pathwidth relates to interval graphs, catwidth relates to what we call catval graphs. We introduce the notion of an extended asteroidal triple (XAT) and characterize catval graphs as the XAT-free chordal graphs. We provide alternative characterizations of these graphs, show that there are graph classes for which the various parameters differ by an arbitrary amount, and consider algorithms for computing catwidth. [Copyright &y& Elsevier]