The connected edge geodetic number of a graph

For a non-trivial connected graph 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 and any edge geodetic set of order g1(G) is an edge geodetic basis. A connected edge geodetic set of G is an edge geodetic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected edge geodetic set of G is the connected edge geodetic number of G and is denoted by g1c(G). A connected edge geodetic set of cardinality g1c(G) is called a g1c set of G or a connected edge geodetic basis of G. Some general properties satisfied by this concept are studied. The connected edge geodetic number of certain classes of graphs are determined. Connected graphs of order p with connected edge geodetic number 2 are characterized. Various necessary conditions for the connected edge geodetic number of a graph to be p − 1 or p are given. It is shown that every pair k, p of integers with 3 6 k 6 p is realizable as the connected edge geodetic number and order of some connected graph. For positive integers r, d and n > d + 1 with r 6 d 6 2r, there exists a connected graph of radius r, diameter d and connected edge geodetic number n. If p, d and n are integers such that 2 6 d 6 p − 1 and d + 1 6 n 6 p, then there exists a connected graph G of order p, diameter d and g1c(G) = n. Also if p, a and b are positive integers such that 2 6 a < b 6 p, then there exists a connected graph G of order p, g1(G) = a and g1c(G) = b.