A characterization of biconnected graphs reachable by robots jumping over m obstacles

Given two vertices u, v in a graph G, by Cvu we denote the configuration of G with a robot at vertex u, a hole at vertex v and obstacles at the remaining vertices of G graph G is said to be complete S-reachable if starting from each configurations in G the robot can be taken to any other vertex of G by a sequence of moves consisting of simple moves of the obstacles and mRJ moves of the robot for m∈S, where S is a finite non-empty set of non-negative integers. An mRJ move on Cvu is the process of moving the robot from u to v by jumping over m obstacles if there is a u-v path of length m+1 in G. A 0RJ move is known as a simple move. We characterize the biconnected graphs that are complete {m}-reachable, for some positive integer m.