The Forcing Edge-to-Vertex Geodetic Number of a Graph

Abstract

For a connected graph G = (V, E), a set 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 g ev (G). Any edge-to-vertex geodetic set of cardinality g ev (G) is called an edge-to-vertex geodetic basis of G. A subset T ⊆ S is called a forcing subset for S if S is the unique minimum edge-to-vertex geodetic set containing T . A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing edge-to-vertex geodetic number of S, denoted by f ev (S), is the cardinality of a minimum forcing subset of S. The forcing edge-to-vertex geodetic number of G, denoted by f ev (G), is f ev (G) = min {f ev (S)}, where the minimum is taken over all minimum edgeto-vertex geodetic sets S in G. Some general properties satisfied by the concept forcing edge-to-vertex geodetic number is studied. The forcing edge-to-vertex geodetic number of certain classes of graphs are determined. It is shown that

The edge-to-edge geodetic domination number of a graph

“…A vertex x is said to lie on an A-B geodesic if x is a vertex of an A-B geodesic [21]. A set S ⊆ E(G) is called an edge-to-vertex geodetic set 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 edge-to-vertex geodetic number g ev (G) of G is the minimum cardinality of its edge-to-vertex geodetic sets and any edge-tovertex geodetic set of cardinality g ev (G) is called an g ev -set of G. The edge-to-vertex geodetic number of a graph was studied in [21,24,25,27]. A set S ⊆ E(G) is called an edge-to-edge geodetic set of G if every edge of G is an element of S or lies on a geodesic joining a pair of edges of S. The edge-to-edge geodetic number g ee (G) of G is the minimum cardinality of its edge-to-edge geodetic sets and any edge-to-edge geodetic set of cardinality g ee (G) is said to be a g ee -set of G. This concept was studied in [1].…”