Completely Strong Path-Connectivity of Local Tournaments

Abstract

Let $T=(V,A)$ be an oriented graph with $n$ vertices. $T$ is completely strong path-connected if for each arc $(a,b)\in A$ and $k$ ($k=2,\dots ,n-1$), 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 $P_k^{\prime }(a,b)$) in $T$. In this paper, we prove that a connected local tournament $T$ is completely strong path-connected if and only if for each arc $(a,b)\in A$, there exist $P_2(a,b)$ and $P_2^{\prime }(a,b)$ in $T$, and $T$ is not of $T_1≆T_0$-$D_8^{\prime }$-type digraph and $D_8$.