The upper edge geodetic number and the forcing edge geodetic number of a graph

An edge geodetic set of a connected graph G of order p ≥ 2 is a set S ⊆ V(G) such that 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 cardinality of its edge geodetic sets and any edge geodetic set of cardinality g1(G) is a minimum edge geodetic set of G or an edge geodetic basis of G. An edge geodetic set S in a connected graph G is a minimal edge geodetic set if no proper subset of S is an edge geodetic set of G. The upper edge geodetic number g1+(G) of G is the maximum cardinality of a minimal edge geodetic set of G. The upper edge geodetic number of certain classes of graphs are determined. It is shown that for every two integers a and b such that 2 ≤ a ≤ b, there exists a connected graph G with g1(G) = a and g1+(G) = b. For an edge geodetic basis S of G, a subset T ⊆ S is called a forcing subset for S if S is the unique edge geodetic basis containing T. A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing edge geodetic number of S denoted by ƒ1(S), is the cardinality of a minimum forcing subset of S. The forcing edge geodetic number of G, denoted by ƒ1(G), is ƒ1(G) = min{ ƒ1(S)}, where the minimum is taken over all edge geodetic bases S in G. Some general properties satisfied by this concept are studied. The forcing edge geodetic number of certain classes of graphs are determined. It is shown that for every pair a, b of integers with 0 ≤ a < b and b ≥ 2, there exists a connected graph G such thatƒ1(G) = a and g1(G) = b.