On the intersection of tolerance and cocomparability graphs

Mertzios, G.B.; Zaks, S.

Abstract

It has been conjectured by Golumbic and Monma in 1984 that the intersection of tolerance and cocomparability graphs coincides with bounded tolerance graphs. Since cocomparability graphs can be efficiently recognized, a positive answer to this conjecture in the general case would enable us to efficiently distinguish between tolerance and bounded tolerance graphs, although it is NP-complete to recognize each of these classes of graphs separately. The conjecture has been proved under some – rather strong – structural assumptions on the input graph; in particular, it has been proved for complements of trees, and later extended to complements of bipartite graphs, and these are the only known results so far. Furthermore, it is known that the intersection of tolerance and cocomparability graphs is contained in the class of trapezoid graphs. In this article we prove that the above conjecture is true for every graph G, whose tolerance representation satisfies a slight assumption; note here that this assumption concerns only the given tolerance representation R of G, rather than any structural property of G. This assumption on the representation is guaranteed by a wide variety of graph classes; for example, our results immediately imply the correctness of the conjecture for complements of triangle-free graphs (which also implies the above-mentioned correctness for complements of bipartite graphs). Our proofs are algorithmic, in the sense that, given a tolerance representation R of a graph G, we describe an algorithm to transform R into a bounded tolerance representation R  ∗  of G. Furthermore, we conjecture that any minimal tolerance graph G that is not a bounded tolerance graph, has a tolerance representation with exactly one unbounded vertex. Our results imply the non-trivial result that, in order to prove the conjecture of Golumbic and Monma, it suffices to prove our conjecture. In addition, there already exists evidence in the literature that our conjecture is true.

Citation

Mertzios, G., & Zaks, S. (2010, December). On the intersection of tolerance and cocomparability graphs. Presented at 21st International Symposium on Algorithms and Computation (ISAAC), Jeju Island, Korea

• Presentation Conference Type: Conference Paper (published)
• Conference Name: 21st International Symposium on Algorithms and Computation (ISAAC)
• Publication Date: Dec 1, 2010
• Deposit Date: Dec 8, 2011
• Publicly Available Date: Mar 6, 2012
• Print ISSN: 0302-9743
• Pages: 230-240
• Series Title: Lecture notes in computer science
• Series Number: 6506
• Series ISSN: 0302-9743,1611-3349
• Book Title: Algorithms and Computation, 21st International Symposium, ISAAC 2010, 15-17 December 2010, Jeju Island, Korea ; proceedings Part I.
• DOI: https://doi.org/10.1007/978-3-642-17517-6_22
• Keywords: Tolerance graphs, Cocomparability graphs, 3-dimensional intersection model, Trapezoid graphs, Parallelogram graphs.
• Public URL: https://durham-repository.worktribe.com/output/1157453
• Additional Information: Proceedings published as: Algorithms and Computation: 21st International Symposium, ISAAC 2010, Jeju Island, Korea, December 15-17, 2010, Proceedings, Part I Event url: http://tclab.kaist.ac.kr/~isaac10/home.html