Graph isomorphism completeness for trapezoid graphs

TL;DR

This paper proves that the graph isomorphism problem is GI-complete for trapezoid graphs, resolving a decade-long open question and delineating the boundary between polynomial-time solvability and GI-completeness.

Contribution

It establishes the GI-completeness of the graph isomorphism problem for trapezoid graphs, specifically for comparability graphs of certain posets, filling a key gap in complexity classification.

Findings

GI-complete for comparability graphs of posets with interval dimension 2 and height 3

Polynomial-time solvable for posets with interval dimension ≤ 2 and height ≤ 2

Clarifies the boundary between tractable and GI-complete cases

Abstract

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.