a characterization of the centers of chordal graphs

Abstract

A graph is $k$-chordal if it does not have an induced cycle with length greater than $k$. We call a graph chordal if it is $3$-chordal. Let $G$ be a graph. The distance between the vertices $x$ and $y$, denoted by $d_{G}(x,y)$, is the length of a shortest path from $x$ to $y$ in $G$. The eccentricity of a vertex $x$ is defined as $\epsilon_{G}(x)= \max\{d_{G}(x,y)|y\in V(G)\}$. The radius of $G$ is defined as $Rad(G)=\min\{\epsilon_{G}(x)|x\in V(G)\}$. The diameter of $G$ is defined as $Diam(G)=\max\{\epsilon_{G}(x)|x\in V(G)\}$. The graph induced by the set of vertices of $G$ with eccentricity equal to the radius is called the center of $G$. In this paper we present new bounds for the diameter of $k$-chordal graphs, and we give a concise characterization of the centers of chordal graphs.