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We study the problem of finding the next-to-shortest paths in a graph. A next-to-shortest (u, v)-path is a shortest (u, v)-path amongst (u, v)-paths with length strictly greater than the length of the shortest (u, v)-path. In constrast to the situation in directed graphs, where the problem has been shown to be NP-hard, providing edges of length zero are… (More)

We show that for any k there is a polynomial time algorithm to evaluate the weighted graph polynomial U of any graph with tree-width at most k at any point. For a graph with n vertices, the algorithm requires O(a k n 2k+3) arithmetical operations , where a k depends only on k.

We consider the equivalence classes of graphs induced by the unsigned versions of the Reidemeister moves on knot diagrams. Any graph which is reducible by some finite sequence of these moves, to a graph with no edges is called a knot graph. We show that the class of knot graphs strictly contains the set of delta-wye graphs. We prove that the dimension of… (More)

It is known that evaluating the Tutte polynomial, T (G; x, y), of a graph, G, is #P-hard at all but eight specific points and one specific curve of the (x, y)-plane. In contrast we show that if k is a fixed constant then for graphs of tree-width at most k there is an algorithm that will evaluate the polynomial at any point using only a linear number of… (More)

We introduce the greedy expectation algorithm for the fixed spectrum version of the frequency assignment problem. This algorithm was previously studied for the travelling salesman problem. We show that the domination number of this algorithm is at least σ n−⌈log 2 n⌉−1 where σ is the available span and n the number of vertices in the constraint graph. In… (More)

Motivated by problems in radio channel assignment, we consider the vertex-labelling of graphs with nonnegative integers. The objective is to minimize the span of the labelling, subject to constraints imposed at graph distances one and two. We show that the minimum span is (up to rounding) a piecewise linear function of the constraints, and give a complete… (More)

We show that the problem of recognising a partitionable simplicial complex is a member of the complexity class N P, thus answering a question raised in [1].

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