Abstract
Tolerance graphs have been extensively studied since their introduction, due to their interesting structure and their numerous applications, as they generalize both interval and permutation graphs in a natural way. It has been conjectured by Golumbic, Monma, and Trotter 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. This longstanding 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. Our main result in this article is that the above conjecture is true for every graph G that admits a tolerance representation with exactly one unbounded vertex; note that this assumption concerns only the given tolerance representation R of G, rather than any structural property of G. Moreover, our results imply as a corollary that the conjecture of Golumbic, Monma, and Trotter is true for every graph G=(V,E) that has no three independent vertices a,b,c∈V such that N(a)⊂N(b)⊂N(c), where N(v) denotes the set of neighbors of a vertex v∈V; this is satisfied in particular when G is the complement of a triangle-free graph (which also implies the above-mentioned correctness for complements of bipartite graphs). Our proofs are constructive, in the sense that, given a tolerance representation R of a graph G, we 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, Monma, and Trotter, it suffices to prove our conjecture.
Highlights
IntroductionFor a brief description of this intersection model we refer to Section 2 (see Fig. 1(a) and (b) for an illustration)
A simple undirected graph G = (V, E) on n vertices is called a tolerance graph if there exists a collection I = {Iu | u ∈ V } of closed intervals on the real line and a set t = {tu | u ∈ V } of positive numbers, such that for any two vertices u, v ∈ V, uv ∈ E if and only if |Iu ∩ Iv| ≥ min{tu, tv}
In the definition we introduce the notion of the right border property of a vertex u in a projection representation R of a tolerance graph G
Summary
For a brief description of this intersection model we refer to Section 2 (see Fig. 1(a) and (b) for an illustration) This parallelepiped representation of tolerance graphs generalizes the parallelogram representation of bounded tolerance graphs; the main idea is to exploit the third dimension to capture the information given by unbounded tolerances. Our main result is that Conjecture 1 is true for every graph G, for which there exists a tolerance representation with exactly one unbounded vertex. Since Condition 3 is weaker than both Conditions 1 and 2, the same result immediately follows by assuming that the graph G satisfies Condition 1 or Condition 2 This immediately implies our main result, i.e. that Conjecture 1 is true for every graph G that admits a tolerance representation with exactly one unbounded vertex (i.e. when Condition 1 is satisfied).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
