Hangable Graphs

Abstract

Let $G=(V_G,E_G)$ be a connected graph. The distance $d_G(u,v)$ between vertices $u$ and $v$ in $G$ is the length of a shortest $u-v$ path in $G$. The eccentricity of a vertex $v$ in $G$ is the integer $e_G(v)= \max\{ d_G(v,u) \colon u\in V_G\}$. The diameter of $G$ is the integer $d(G)= \max\{e_G(v)\colon v\in V_G\}$. The periphery of a~vertex $v$ of $G$ is the set $P_G(v)= \{u\in V_G\colon d_G(v,u)= e_G(v)\}$, while the periphery of $G$ is the set $P(G)= \{v\in V_G\colon e_G(v)=d(G)\}$. We say that graph $G$ is hangable if $P_G(v)\subequal P(G)$ for every vertex $v$ of $G$. In this paper we prove that every block graph is hangable and discuss the hangability of products of graphs.