Graph isomorphism completeness for trapezoid graphs

The complexity of the graph isomorphism problem for trapezoid graphs has been open over a decade. This paper shows that the problem is GI-complete. More precisely, we show that the graph isomorphism problem is GI-complete for comparability graphs of partially ordered sets with interval dimension 2 and height 3. In contrast, the problem is known to be solvable in polynomial time for comparability graphs of partially ordered sets with interval dimension at most 2 and height at most 2.Comment: 4 pages, 3 Postscript figure

On the intersection of tolerance and cocomparability graphs.

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

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum