An algorithm and its role in the study of optimal graph realizations of distance matrices

AbstractThe algorithm we present is a natural next step to well-known algorithms for finding optimal graph realizations of tree-realizable distance matrices. It is based on the fact, which we prove first, that the quest for optimal realizations of nontree-realizable distance matrices can be narrowed to a proper subclass D∗ of the class D of all such matrices. The matrices in D∗ are those which satisfy the following condition: for each pair of indices {h, i}, there is another pair {j, k} such that the submatrix 〈{h, i, j, k}〉 is nontree-realizable. Given an arbitrary distance matrix D in D, the algorithm associates to D a matrix D∗ in the subclass D∗, whose optimal realization, if known, easily yields the optimal realization of D. The practical usefulness of this algorithm is underscored by a growing number of distance matrices whose optimal realizations are known [4, 7]. Time and space requirements of the algorithm are also discussed

Subgraphs as circuits and bases of matroids

AbstractAs is well known, the cycles of any given graph G may be regarded as the circuits of a matroid defined on the edge set of G. The question of whether other families of connected graphs exist such that, given any graph G, the subgraphs of G isomorphic to some member of the family may be regarded as the circuits of a matroid defined on the edge set of G led us, in two other papers, to the proof of some results concerning properties of the cycles when regarded as circuits of such matroids. Here we prove that the wheels share many of these properties with the cycles. Moreover, properties of subgraphs which may be regarded as bases of such matroids are also investigated

Joins of n-degenerate graphs and uniquely (m, n)-partitionable graphs

AbstractThe Lick-White point-partition numbers generalize the chromatic number and the point-arboricity. Similarly, uniquely (m, n)-partitionable graphs generalize uniquely m-colorable graphs.Theorem 1 gives a method for constructing uniquely (m, n)-partitionable graphs as well as a sufficient condition for a join of m n-degenerate graphs to be uniquely (m, n)-partitionable. For the case n = 1, we obtain a necessary and sufficient condition (Lemma 1). As a consequence, an embedding result for uniquely (m, 1)-partitionable graphs is obtained (Theorem 2). Finally, uniquely (m, n)-partitionable graphs with minimal connectivity are constructed (Theorem 3)

A note on distance matrices with unicyclic graph realizations

AbstractWe give necessary and sufficient conditions for a distance matrix to have a unicyclic graph as unique optimal graph realization