Steven D. Noble

Evaluating the Tutte Polynomial for Graphs of Bounded Tree-Width

It is known that evaluating the Tutte polynomial, T(G; x, y), of a graph, G, is nP-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 multiplications and additions.

The clustering coefficient of a scale-free random graph

We consider a random graph process in which, at each time step, a new vertex is added with m out-neighbours, chosen with probabilities proportional to their degree plus a strictly positive constant. We show that the expectation of the clustering coefficient of the graph process is asymptotically proportional to lognn. Bollobas and Riordan have previously shown that when the constant is zero, the same expectation is asymptotically proportional to (logn)^2n.

Finding next-to-shortest paths in a graph

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 contrast to the situation in directed graphs, where the problem has been shown to be NP-hard, providing edges of length zero are allowed, we prove the somewhat surprising result that there is a polynomial time algorithm for the undirected version of the problem.

Domination analysis of greedy heuristics for the frequency assignment problem

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 ?n-?log2n?-1, where ? is the available span and n the number of vertices in the constraint graph. In contrast to this we show that the standard greedy algorithm has domination number strictly less than ?ne-5(n-1)/144 for large n and fixed ?.

On the structure of the h-vector of a paving matroid

We give two proofs that the h-vector of any paving matroid is a pure O-sequence, thus answering in the affirmative a conjecture made by Stanley, for this particular class of matroids. We also investigate the problem of obtaining good lower bounds for the number of bases of a paving matroid given its rank and number of elements.

The Equivalence of Two Graph Polynomials And a Symmetric Function

The U-polynomial, the polychromate and the symmetric function generalization of the Tutte polynomial, due to Stanley, are known to be equivalent in the sense that the coefficients of any one of them can be obtained as a function of the coefficients of any other. The definition of each of these functions suggests a natural way in which to strengthen them, which also captures Tuttes universal V-function as a specialization. We show that the equivalence remains true for the strong functions, thus answering a question raised by Dominic Welsh.

Minimizing the Oriented Diameter of a Planar Graph

We consider the problem of minimizing the diameter of an orientation of a planar graph. A result of Chvatal and Thomassen shows that for general graphs, it is NP- complete to decide whether a graph can be oriented so that its diameter is at most two. In contrast to this, for each constant l, we describe an algorithm that decides if a planar graph G has an orientation with diameter at most l and runs in time O(c|V |), where c depends on l.

Optimal arrangement of data in a tree directory

We define the decision problem DATA ARRANGEMENT, which involves arranging the vertices of a graph G at the leaves of a d-ary tree so that a weighted sum of the distances between pairs of vertices measured with respect to the tree topology is at most a given value. We show that DATA ARRANGEMENT is strongly NP-complete for any fixed d ≥ 2 and explain the connection between DATA ARRANGEMENT and arranging data in a particular form of distributed directory.

Recognising a partitionable simplicial complex is in NP

Abstract We show that the problem of recognising a partitionable simplicial complex is a member of the complexity class NP, thus answering a question raised by Kleinschmidt and Onn (1995).

The Merino-Welsh conjecture holds for series-parallel graphs

The Merino-Welsh conjecture asserts that the number of spanning trees of a graph is no greater than the maximum of the numbers of totally cyclic orientations and acyclic orientations of that graph. We prove this conjecture for the class of series-parallel graphs.