Graphs with large total geodetic number

Abstract

For two vertices u and v of a graph G, the set I[u, v] consists of all vertices lying on some u − v geodesic in G. If S is a set of vertices of G, then I[S] is the union of all sets I[u, v] for u, v ∈ S. A set of vertices S ⊆ V(G) is a total geodetic set if I[S] = V(G) and the subgraph G[S] induced by S has no isolated vertex. The total geodetic number, denoted by t (G), is the minimum cardinality among all total geodetic sets of G. In this paper, we characterize all connected graphs G of order n ≥ 3 with t (G) = n − 1.