# The i-chords of cycles and paths

• Volume: 32, Issue: 4, page 607-615
• ISSN: 2083-5892

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## Abstract

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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.

## How to cite

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Terry A. McKee. "The i-chords of cycles and paths." Discussiones Mathematicae Graph Theory 32.4 (2012): 607-615. <http://eudml.org/doc/270941>.

@article{TerryA2012,
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.},
author = {Terry A. McKee},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {chord; chordal graph; strongly chordal graph; ptolemaic graph; trivially perfect graph; threshold graph; Ptolemaic graph},
language = {eng},
number = {4},
pages = {607-615},
title = {The i-chords of cycles and paths},
url = {http://eudml.org/doc/270941},
volume = {32},
year = {2012},
}

TY - JOUR
AU - Terry A. McKee
TI - The i-chords of cycles and paths
JO - Discussiones Mathematicae Graph Theory
PY - 2012
VL - 32
IS - 4
SP - 607
EP - 615
AB - 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.
LA - eng
KW - chord; chordal graph; strongly chordal graph; ptolemaic graph; trivially perfect graph; threshold graph; Ptolemaic graph
UR - http://eudml.org/doc/270941
ER -

## References

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5. [5] E. Howorka, A characterization of ptolemaic graphs, J. Graph Theory 5 (1981) 323-331, doi: 10.1002/jgt.3190050314. Zbl0437.05046
6. [6] J. Liu and H.S. Zhou, Dominating subgraphs in graphs with some forbidden structures, Discrete Math. 135 (1994) 163-168, doi: 10.1016/0012-365X(93)E0111-G. Zbl0812.05052
7. [7] N.V.R. Mahadev and U.N. Peled, Threshold Graphs and Related Topics (North-Holland, Amsterdam, 1995). Zbl0852.05001
8. [8] A. McKee, Constrained chords in strongly chordal and distance-hereditary graphs, Utilitas Math. 87 (2012) 3-12. Zbl1264.05112
9. [9] T.A. McKee and F.R. McMorris, Topics in Intersection Graph Theory (Society for Industrial and Applied Mathematics, Philadelphia, 1999). Zbl0945.05003
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