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Publications

# A characterization of the Centers of Chordal Graphs

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**Date Published:** September 2022

#### Author(s)

James Shook (NIST), Bing Wei (University of Mississippi)

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.

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.

Hide full abstract
#### Keywords

graph theory; chordal graph; center; diameter
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