Completely strong path-connected tournaments

Abstract Let T = ( V , A ) be a tournament with p vertices. T is called completely strong path-connected if for each arc ( a , b ) ∈ A and k ( k = 2, 3,…, p ), there is a path from b to a of length k (denoted by P k ( a , b )) and a path from a to b of length k (denoted by P ′ k ( a , b )). In this paper, we prove that T is completely strong path-connected if and only if for each arc ( a , b ) ∈ A , there exist P 2 ( a , b ), P ′ 2 ( a , b ) in T , and T satisfies one of the following conditions: (a) T / T 0 -type graph, (b) T is 2-connected, (c) for each arc ( a , b ) ∈ A , there exists a P ′ p −1 ( a , b ) in T .