FORCING TOLL CONVEXITY NUMBERS OF SOME PRODUCTS OF GRAPHS

Let G be a connected graph and let u,v is an element of V(G) A tolled walk T between u and v in G is a walk of the form T : u, w(1),..., w(k), u, where k >= 1, in which w(1) and w(2) are the only neighbors of u and v in T, respectively. The toll interval T-G(u, v) is defined as the set of vertices in G that lie on some u - v walk. A subset S subset of V(G) is toll convex (or t-convex) if T-G (u, v) subset of S for all u, v is an element of S. A subset R of a maximum t-convex set or a con(T)- set S of G is called a forcing subset for S if S is the unique con(T)- set containing R. The forcing toll convexity number f con(T)(S) of a con(T)- set S of G is the minimum cardinality of a forcing subset for S. The forcing toll convexity number f con(T)(G) of G is the minimum forcing toll convexity number among all con(T)- sets of G. In this paper we characterize the toll convex sets and con(T)- sets in the Cartesian and lexicographic product of some graphs. Moreover, we determine the forcing toll convexity numbers of these products of graphs.