Discussiones Mathematicae Graph Theory 32(4) (2012) 607-615
doi: 10.7151/dmgt.1629

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The i-chords of Cycles and Paths

Terry A. McKee

Department of Mathematics and Statistics
Wright State University
Dayton, Ohio 45435 USA

Abstract

An i-chord of a cycle or path is an edge whose endpoints are a distance i ≥ 2 apart along the cycle or path. Motivated by many standard graph classes being describable by the existence of chords, we investigate what happens when i-chords are required for specific values of i. Results include the following: A graph is strongly chordal if and only if, for i ∈ {4,6}, every cycle C with |V(C) | ≥ i has an (i/2)-chord. A graph is a threshold graph if and only if, for i ∈ {4,5}, every path P with |V(P) | ≥ i has an (i −2)-chord.

Keywords: chord, chordal graph, strongly chordal graph, ptolemaic graph, trivially perfect graph, threshold graph

2010 Mathematics Subject Classification: 05C75, 05C38.

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Received 29 July 2011
Revised 4 November 2011
Accepted 4 November 2011