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2 - Early work on tolerance graphs

Published online by Cambridge University Press:  11 August 2009

Martin Charles Golumbic
Affiliation:
University of Haifa, Israel
Ann N. Trenk
Affiliation:
Wellesley College, Massachusetts
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Summary

In this chapter we discuss the definitions, results and open questions that appear in Golumbic and Monma (1982) and Golumbic, Monma, and Trotter (1984), the papers which introduced the topic of tolerance graphs. We also present consequences of these results and related topics from more recent literature.

Notation and observations

Recall from Section 1.3 that a graph G is a tolerance graph if each vertex νV(G) can be assigned a closed interval Iν and a tolerance tνR+ so that xyE(G) if and only if |IxIy| ≥ min{tx, ty}. If graph G has a tolerance representation with tν ≤ |Iν| for all νV(G), then G is called a bounded tolerance graph.

Many important graph properties are inherited by all induced subgraphs and thus called hereditary properties. Given a (bounded) tolerance representation 〈I, t〉 of a graph G, for any subset of vertices WV(G) the intervals {Iw | wW} and tolerances {tw | wW} give a representation of GW. Thus, induced subgraphs of tolerance graphs are also tolerance graphs and induced subgraphs of bounded tolerance graphs are also bounded tolerance graphs. We record this as a remark.

Remark 2.1. The property of being a tolerance graph (resp. bounded tolerance graph) is hereditary.

In a tolerance representation of a graph G, we may have intervals of the form Ix = [ax, ax].

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Tolerance Graphs , pp. 29 - 52
Publisher: Cambridge University Press
Print publication year: 2004

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