Two edge colorings of a graph are edge-Kempe equivalent if one can be obtained from the other by a series of edge-Kempe switches. This work gives some results for the number of edge-Kempe equivalence classes for cubic graphs. In particular we show every 2-connected planar bipartite cubic graph has exactly one edge-Kempe equivalence class. Additionally, we exhibit infinite families of nonplanar ...

Given a complete graph Kn = (V,E) with non-negative edge costs c ∈ R , the problem multi-2ECcost is that of finding a 2-edge connected spanning multi-subgraph of Kn with minimum cost. It is believed that there are no efficient ways to solve the problem exactly, as it is NP-hard. Methods such as approximation algorithms, which rely on lower bounds like the linear programming relaxation multi-2EC...

For a nontrivial connected graph G = (V (G), E(G)), a set S ⊆ V (G) is called an edge geodetic set of G if every edge of G is contained in a geodesic joining some pair of vertices in S. The edge geodetic number g1(G) of G is the minimum order of its edge geodetic sets. Bounds for the edge geodetic number of Cartesian product graphs are proved and improved upper bounds are determined for a speci...

Directed graphs are widely used in modelling of nonsymmetric relations in various sciences and engineering disciplines. We discuss invariants of strongly connected directed graphs minimal number of vertices or edges necessary to remove to make remaining graphs not strongly connected. By analogy with undirected graphs these invariants are called strong vertex/edge connectivities. We review first...

A necessary and sufficient condition for a graph to be minimally k-edge-connected where k is odd is presented. Based on this result, a divide-and-conquer algorithm to determine minimally k-edge-connected graphs is developed. Experiments are performed to check minimally 3-edge-connectivity. The results show that our algorithm is much more effective than the previously known best algorithm.

A digraph G = (V, E) with diameter D is said to be s-geodetic, for 1 ≤ s ≤ D, if between any pair of (not necessarily different) vertices x, y ∈ V there is at most one x → y path of length ≤ s. Thus, any loopless digraph is at least 1-geodetic. A similar definition applies for a graph G, but in this case the concept is closely related to its girth g, for then G is s-geodetic with s = b(g − 1)/2...

In this work, first, we present sufficient conditions for a bipartite digraph to attain optimum values of a stronger measure of connectivity, the so-called superconnectivity. To be more precise, we study the problem of disconnecting a maximally connected bipartite (di)graph by removing nontrivial subsets of vertices or edges. Within this framework, both an upper-bound on the diameter and Chartr...

We study two problems. In correlation clustering, the input is a weighted graph, where every edge labelled either $$\langle +\rangle $$ or -\rangle according to whether its endpoints are in same category different categories. The goal produce partition of vertices into categories that tries respect labels edges. two-edge-connected augmentation, graph and subset R edges graph. minimum weight S s...