The Forcing Triangle Free Detour Convexity Number of a Graph

Abstract

Let  be a connected graph and  be a -convex set of  A subset   is called a forcing subset for   if is the unique -convex set containing  A forcing subset for  of minimum cardinality is a minimum forcing subset of . The forcing triangle free detour convexity number () of  is the minimum cardinality among the forcing subset for  and the forcing triangle free detour convexity number )  of  is the minimum forcing triangle free detour convexity number among all triangle free detour convex set of  Some general properties satisfied by this concept are studied. The forcing  triangle free detour convexity numbers of certain classes of graphs are determined. It is shown for every pair  of integers with  and , there exists a connected graph  such that  and