The upper 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-setof G or connected edge geodetic basis of G. A connected edge geodetic set S in a connected graph G is called a minimal connected edge geodetic set if no proper subset of S is a connected edge geodetic set of G. The upper connected edge geodetic number g 1 c + ( G ) is the maximum cardinality of a minimal connected edge geodetic set of G. Graphs G of order p for which g 1 c ( G ) = g 1 c + = p are characterized. For positive integers r,d and n ≥ d + 1 with r ≤ d ≤ 2r, there exists a connected graph of radius r, diameter d and upper connected edge geodetic number n. It is shown for any positive integers 2 ≤ a < b ≤ c, there exists a connected graph G such that g1(G) = a,g1c(G) = b and g 1 c + ( G ) = c .