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Abstract: We introduce the process number of a digraph as a tool to study rerouting issues in wdm networks. This parameter is closely related to the vertex separation (or pathwidth). We consider the recognition and the characterization of (di)graphs with small process numbers. In particular, we give a linear time algorithm to recognize (and process) graphs with process number at most 2, along with a characterization in terms of forbidden minors, and a structural description. As for digraphs with process number 2, we exhibit a characterization that allows one to recognize (and process) them in polynomial time. [Copyright &y& Elsevier]

Abstract: We study the bandwidth and the pathwidth of multi-dimensional grids. It can be shown for grids, that these two parameters are equal to a more basic graph parameter, the vertex boundary width. Using this fact, we determine the bandwidth and the pathwidth of three-dimensional grids, which were known only for the cubic case. As a by-product, we also determine the two parameters of multi-dimensional grids with relatively large maximum factors. [Copyright &y& Elsevier]

We describe an O(nlogn) algorithm for the computation of the vertex separation of unicyclic graphs. The algorithm also computes a linear layout with optimal vertex separation in the same time bound. Pathwidth, node search number and vertex separation are different ways of defining the same notion. Path decompositions and search strategies can be derived from linear layouts. The algorithm applies existing, linear time, techniques for the computation of the vertex separation of trees together with corresponding optimal layouts. We reformulate the earlier work on the linear time computation of optimal layouts. A polynomial time algorithm for the problem on unicyclic graphs can be inferred from existing more general methods for graphs of fixed treewidth, since unicyclic graphs have treewidth two, but the time complexity of the resulting method would seem to be an inordinately high order polynomial. Our algorithm we claim is “practical.” The addition of one edge to a tree does seem to require a considerably more elaborate algorithm. [Copyright &y& Elsevier]

The interval degree id(G) of a graph G is defined to be the smallest max-degree of any interval supergraphs of G. One of the reasons for our interest in this parameter is that the bandwidth of a graph is always between id(G)/2 and id(G). We prove also that for any graph G the interval degree of G is at least the pathwidth of G2. We show that if G is an AT-free claw-free graph, then the interval degree of G is equal to the clique number of G2 minus one. Finally, we show that there is a polynomial time algorithm for computing the interval degree of AT-free claw-free graphs. [Copyright &y& Elsevier]