The Closed Geodetic Numbers ofGrap

摘要

tudy on the closed geodetic number evolved from two classes of graphical games called achievement and avoidance games [10], Let G = (V(G),E(G)) be a connected graph. For S C V(G), the geodetic closure IQ[S] of S is the set of all vertices on geodesies (shortest paths) between two vertices of S. We select vertices of G sequentially as follows: Select a vertex v_1. and let S_1={v_1} Select a vertex S_2={v_1,v_2}. Then successively select vertex v_i ∈IG[S_(i-1)] and let S_i={jv_1,v_2,...,v_i}. Since the set V(G) is finite, there is & k for which IciSk] = V(G). We define the closed geodetic number of G, denoted cgn(G), to be the smallest k for which the selection of v_k in the given manner makes I_G[S_k]=V(G).