A characterization of the Centers of Chordal Graphs

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... See full 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 $ϵ_G(x)=$ max$\{d_G(x,y)|y∈V(G)\}$. The radius of $G$ is defined as $Rad(G)=$ min$\{ϵ_G(x)|x∈V(G)\}$. The diameter of $G$ is defined as $Diam(G)=$ max$\{ϵ_G(x)|x∈V(G)\}$. 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.
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