The Connected Edge-To-Vertex Geodetic Number of a Graph

Abstract

Let G = (V, E) be a graph. A subset S ⊆ E is called an edge-to-vertex geodetic set of G if every vertex of G is either incident with an edge of S or lies on a geodesic joining a pair of edges of S. The minimum cardinality of an edge-to-vertex geodetic set of G is gev(G). Any edge-to-vertex geodetic set of cardinality gev(G) is called an edge-to-vertex geodetic basis of G. A connected edge-to-vertex geodetic set of a graph G is an edge-to-vertex geodetic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected edge-to-vertex geodetic set of G is the connected edge-to-vertex geodetic number of G and is denoted by gcev(G). Some general properties satisfied by this concept are studied. The connected graphs G of size q with connected edge-to-vertex geodetic number 2 or q or q − 1 are characterized. It is shown that for any three positive integers q, a and b with 2 ≤ a ≤ b ≤ q, there exists a connected graph G of size q, gev(G) = a and gcev(G) = b.