Dominating vertex covers: the vertex-edge domination problem

Abstract

The vertex-edge domination number of a graph, γ ve (G), is defined to be the cardinality of a smallest set D such that there exists a vertex cover C of G such that each vertex in C is dominated by a vertex in D. This is motivated by the problem of determining how many guards are needed in a graph so that a searchlight can be shone down each edge by a guard either incident to that edge or at most distance one from a vertex incident to the edge. Our main result is that for any cubic graph G with n vertices, γ ve (G) ≤ 9n/26. We also show that it is N P-hard to decide if γ ve (G) = γ(G) for bipartite graph G.

A New Direct Coefficient-Based Heuristic Algorithm for Set Covering Problems

“…The set covering problem has an exponential complexity and is one of the NP-hard problems [60]. Hence, it is almost impossible to find an optimal solution in polynomial time [61].…”

Vertex cover and Edge vertex domination in trees

“…Relating vertex covering number with other dominating parameters is studied in [1,2]. In this paper, we characterize trees with equal vertex covering number and edge-vertex domination number.…”