altan derivatives of polycyclic conjugated hydrocarbons were recently introduced and studied in theoretical organic chemistry. we now provide a generalization of the altan concept, applicable to any graph. several earlier noticed topological properties of altan derivatives of polycyclic conjugated hydrocarbons are shown to be the properties of all altan derivatives of all graphs. among these ar...

The perfect matching polytope of a graph G is the convex hull of the incidence vectors of all perfect matchings in G. We characterise bipartite graphs and near-bipartite graphs whose perfect matching polytopes have diameter 1.

We give a Cayley type formula to count the number of spanning trees in the complete r-uniform hypergraph for all r ≥ 3. Similar to the bijection between spanning trees of the complete graph on (n + 1) vertices and Parking functions of length n, we derive a bijection from spanning trees of the complete (r + 1)-uniform hypergraph which arise from a fixed r-perfect matching (see Section 2) and r-P...

The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. A connected graph G is a cactus if any two of its cycles have at most one common vertex. Let G (n, r) be the set of cacti of order n and with r cycles, ξ(2n, r) the set of cacti of order 2n with a perfect matching and r cycles. In this paper, we give the sharp upper bounds of t...

The matching preclusion number of a graph is the minimum number of neither edges whose deletion in a graph has a neither perfect matching nor an almost perfect matching. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. Recently the conditional matching preclusion number of a graph was introduced to look for sets beyond those induced by a single...

We show that the visibility graph of a set of non-intersecting translates of the same compact convex object in R always contains a Hamiltonian path. Furthermore, we show that every other edge in the Hamiltonian path can be used to obtain a perfect matching that is realized by a set of non-intersecting lines of sight. ? 2001 Elsevier Science B.V. All rights reserved.

In 1891, Peterson [6] proved that every 3-regular bridge-less graph has a perfect matching. It is well known that the dual of a triangular mesh on a compact manifold is a 3-regular graph. M. Gopi and D. Eppstein [4] use Peterson’s theorem to solve the problem of constructing strips of triangles from triangular meshes on a compact manifold. P. Diaz-Gutierrez and M. Gopi [3] elaborate on the crea...

The matching preclusion problem, introduced by Brigham et al. [Perfect-matching preclusion, Congressus Numerantium 174 (2005) 185-192], studies how to effectively make a graph have neither perfect matchings nor almost perfect matchings by deleting as small a number of edges as possible. Extending this concept, we consider a more general matching preclusion problem, called the strong matching pr...

For planar graphs, counting the number of perfect matchings (and hence determining whether there exists a perfect matching) can be done in NC [4, 10]. For planar bipartite graphs, finding a perfect matching when one exists can also be done in NC [8, 7]. However in general planar graphs (when the bipartite condition is removed), no NC algorithm for constructing a perfect matching is known. We ad...

We give a Cayley type formula to count the number of spanning trees in the complete r-uniform hypergraph for all r ≥ 3. Similar to the bijection between spanning trees in complete graphs and Parking functions, we derive a bijection from spanning trees of the complete (r + 1)-uniform hypergraph which arise from a fixed r-perfect matching (see Section 2) and r-Parking functions. We observe a simp...