The Restrained Edge Geodetic Number of a Graph

A set S of vertices of a connected graph G is a geodetic set if every vertex of G lies on an x−y geodesic for some elements x and y in S. The minimum cardinality of a geodetic set ofG is the geodetic number ofG, denoted by g(G). A set S of vertices of a graph G is an edge geodetic set if every edge of G lies on an x − y geodesic for some elements x and y in S. The minimum cardinality of an edge geodetic set of G is the edge geodetic number of G, denoted by eg(G). A set S of vertices of a graph G is a restrained edge geodetic set if S is an edge geodetic set, and if either V = S or the subgraph G[V − S] induced by V − S has no isolated vertices. The minimum cardinality of a restrained edge geodetic set of G is the restrained edge geodetic number of G and is denoted by egr(G). The restrained edge geodetic numbers of some standard graphs are determined. Some special classes of graphs of order p with egr(G) = p are characterized. It is proved that, for the integers a, b, c and p such that 2 ≤ a ≤ b ≤ c ≤ p with p − c − a − 2 ≥ 0, there exists a connected graph G of order p, geodetic number a, edge geodetic number b and the restrained edge geodetic number c except the values (a, b) ∈ {(2, 2), (2, 3), (3, 4)}. It is also proved that If a, b, c and p are integers such that 3 ≤ a ≤ b ≤ c ≤ p− 3, then there exists a connected graph G of order p, geodetic number a, restrained geodetic number b and the restrained edge geodetic number c. AMS subject classification: 05C12.