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

Let G = (V, E) be a connected graph with at least three vertices. A set S Í E is called an edge-to-edge geodetic dominating set of G if S is both an edge-to-edge geodetic set of G and an edge dominating set of G. The edge-to- edge geodetic domination number ¡gee(G) of G is the minimum cardinality of its edge-to-edge geodetic dominating sets and any edge-to-edge geodetic dominating set of minimum cardinality is said to be a gee- set of G. Some general properties satisfied by this concept are studied. Connected graphs of size m?2 with edge-to-geodetic domination number 2 or m or m-1 are charaterized. We proved that if G is a connected graph of size m ? 3 and G­ is also connected,then 4 ?¡gee(G) + ¡gee(G­) ? 2m -2. Moreover we characterized graphs for which the lower and the upper bounds are sharp. It is shown that, for every pair of positive integers a and b with 2 ?a ? b, there exists a connected graph G with gee(G) = a and ¡gee(G) = b. Also it is shown that, for every pair of positive integers a and b with 2 < a ? b, there exists a connected graph G with ¡e(G) = a and¡ gee(G) = b, where ¡e(G) is the edge domination number of G and gee(G) is the edge-to-edge geodetic number of G.