Hangable Graphs

TL;DR

This paper introduces the concept of hangable graphs, characterizes block graphs as always hangable, and explores how graph products influence hangability, expanding understanding of graph periphery properties.

Contribution

It defines hangable graphs, proves all block graphs are hangable, and analyzes hangability in graph products, providing new insights into graph periphery structures.

Findings

Every block graph is hangable.

The hangability of graph products is characterized.

Properties of peripheries in different graph classes are analyzed.

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.