Let G be a connected graph. Given any two vertices u and v of G, the set  consists of all those vertices lying on a longest u-v path. A set S is a detour convex set if  for  A tolled walk T between distinct vertices u and v of G is a walk of the form  where  in which  and  are the only neighbors of u and v in T, respectively. The toll interval  is the set of vertices in G that lie on some u-v walk. A subset  is toll convex (or t-convex) if  for all

In this paper, we define and study the concepts of detour convexity number, toll convexity number, forcing subset for a maximum detour convex (maximum toll convex) set, and the forcing detour convexity (forcing toll convexity) number of a graph. In particular, we study these concepts in the join and corona of graphs.

detour convex set, toll convex set, forcing detour convexity number, forcing toll convexity number.