In this paper, we describe the use of Riemannian geometry and graph-spectral methods for purposes of minimum spanning tree recovery. We commence by showing how the sectional curvature can be used to model the edge-weights of the graph as a dynamic system in a manifold governed by a Jacobi field. With this characterisation of the edge-weights at hand, we proceed to recover an approximation for t...

An exclusion region for a triangulation is a region that can be placed around each edge of the triangulation such that the region can not contain points from the set on both sides of the edge. We survey known exclusion regions for several classes of triangulations, including Delaunay, Greedy, and Minimum Weight triangulations. We then show an exclusion region of larger area than was previously ...

Abstract. A skew edge coloring of a graph G is defined to be a set of two edge colorings such that no two edges are assigned the same unordered pair of colors. The skew chromatic index s(G) is the minimum number of colors required for a skew edge coloring of G. In this paper, an algorithm is determined for skew edge coloring of circular ladder graphs. Alsothe skew chromatic index of circular la...

In the Perfect Completion problem one wishes to add the fewest possible edges to a graph in order to obtain a perfect graph. How large can the size of the added edge set be compared to the size of the edge set of the original graph? We show that with high probability the smallest perfect graph containing a random graph G(n; n 0:6) has (n 1:8) edges. We also show that with high probability the m...

A skew edge coloring of a graph G is defined to be a set of two edge colorings such that no two edges are assigned the same unordered pair of colors. The skew chromatic index s(G) is the minimum number of colors required for a skew edge coloring of G. In this paper, skew edge coloring of certain classes of graphs are determined. Furthermore, the skew chromatic index of those graphs is obtained ...

We study the graphs G for which the hull number h(G) and the geodetic number g(G) with respect to P3-convexity coincide. These two parameters correspond to the minimum cardinality of a set U of vertices of G such that the simple expansion process that iteratively adds to U , all vertices outside of U that have two neighbors in U , produces the whole vertex set of G either eventually or after on...

Given a directed graph G, a feedback arc set of G is a subset of its edges containing at least one edge of every cycle in G. Finding a feedback arc set of minimum cardinality is the minimum feedback arc set problem. The minimum set cover formulation of the minimum feedback arc set problem is appropriate as long as all the simple cycles in G can be enumerated. Unfortunately, even sparse graphs c...