The Total Geodetic Number of a Graph

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 is an element of S. A set of vertices S subset of V(G) is a geodetic set if I [S] = V(G), and the minimum cardinality of a geodetic set is the geodetic number g(G). A geodetic set S subset of V(G) is a total geodetic set if the subgraph G[S] induced by S has no isolated vertices. The minimum cardinality of a total geodetic set is the total geodetic number gt(G). A subset T of a minimum total geodetic set S is called a forcing subset for S if S is the unique minimum total geodetic set containing T. The forcing total geodetic number of S is the minimum cardinality among the forcing subsets of S, and the forcing total geodetic number (G, gt) of G is the minimum forcing total geodetic number among all minimum total geodetic sets of G. We determine sharp bounds for gt(G) and f (G, gt). The minimum cardinality of a geodetic set which induces a connected graph is the connected geodetic number gc(G). We present necessary and sufficient conditions which determine for every triple of integers a, b, c whether there exists a nontrivial connected graph G with: (i) a = g(G), b = g(t)(G), c = vertical bar V (G)vertical bar; (ii) a = rad G, b = diam G, c = g(t)(G); (iii) a = g(t)(G), b = g(e)(G), c = vertical bar V(G)vertical bar. We find all ordered pairs (a, b) of integers which are realizable as the forcing total geodetic number and total geodetic number for some nontrivial connected graph, respectively.