J. M. S. Simões-Pereira

On optimal embeddings of metrics in graphs

Abstract This paper extends previous results of the authors. In particular, non-treerealizable metrics are investigated and it is shown that every finite metric has an optimal realization by a graph.

On matroids on edge sets of graphs with connected subgraphs as circuits II

Matroidal families are defined as families of connected graphs such that, given any graph G, the subgraphs of G isomorphic to a member of the family are the circuits of a matroid on the edge set of G. It will be proved that there are four matroidal families with all members having less than three independent cycles (Theorems 1 and 2) and that all members of any other matroidal family have at least three independent cycles (Theorem 3). The members of these four families are the complete graph on two points, the cycles, the connected graphs with two independent cycles and no pendant edges (which we call the bicircular graphs), and the members of a family formed by the even cycles and the bicircular graphs with no even cycle, respectively. Concerning any other matroidal families, we prove that no graph in such a family has a vertex of degree two (Theorem 5) and consequently that no two graphs in such a family are homeomorphic (Theorem 6).

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

The 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.

An optimality criterion for graph embeddings of metrics

A criterion for optimality of graph realizations of distance matrices is presented. In order to illustrate the criterion’s potential, it is used to prove the optimality of several well-known graph realizations and a few new ones.

A note on distance matrices with unicyclic graph realizations

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

Subgraphs as circuits and bases of matroids

As 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

Abstract The 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 convexity and submatrices of distance matrices

We prove that, if the index set of a distance matrix cannot be partitioned into two convex subsets, then, given a pair of indices, we can find another pair such that the principal submatrix of order four associated with the four indices is nontree-realizable.

Just two total graphs are complementary

It is proved that there are just two non-trivial total graphs which are complementary, namely the triangle and its complement.