The edge fixed geodomination number of a graph

For a vertex x in a connected graph G = (V (G), E(G)) of order p ≥ 3, a set S ⊆ V (G) is an x-geodominating set of G if each vertex v ∈ V (G) lies on an x-y geodesic for some element y in S. The minimum cardinality of an x-geodominating set of G is defined as the x-geodomination number of G, denoted by gx(G). An x-geodominating set of cardinality gx(G) is called a gx-set of G. For an edge e = xy in G, a set S ⊆ V (G) is an e-geodominating set of G if each vertex v ∈ V (G) lies on either an x − z geodesic or an y − z geodesic for some element z in S. The minimum cardinality of an e-geodominating set of G is defined as the e-geodomination number of G, denoted by ge(G). An e-geodominating set of cardinality ge(G) is called a ge-set of G. Some general properties satisfied by e-geodominating sets are studied. We determine bounds for the e-geodomination number and find the same for some special classes of graphs. For positive integers r, d and n ≥ 2 with r < d ≤ 2r, there exists a connected graph G with rad G = r, diam G = d and gxy(G) = n or n − 1 for any edge xy in G. If p, d and n are integers such that 3 ≤ d ≤ p − 1, 2 ≤ n ≤ p − 2 and p − d − n + 1 ≥ 0, then there exists a graph G of order p, diameter d and gxy(G) = n or n − 1 for any edge xy in G.