The Recognition of Tolerance and Bounded Tolerance Graphs

Tolerance graphs model interval relations in such a way that intervals can tolerate a certain degree of overlap without being in conflict. This subclass of perfect graphs has been extensively studied, due to both its interesting structure and its numerous applications. Several efficient algorithms for optimization problems that are NP-hard on general graphs have been designed for tolerance graphs. In spite of this, the recognition of tolerance graphs - namely, the problem of deciding whether a given graph is a tolerance graph - as well as the recognition of their main subclass of bounded tolerance graphs, have been the most fundamental open problems on this class of graphs (cf. the book on tolerance graphs \cite{GolTol04}) since their introduction in 1982 \cite{GoMo82}. In this article we prove that both recognition problems are NP-complete, even in the case where the input graph is a trapezoid graph. The presented results are surprising because, on the one hand, most subclasses of perfect graphs admit polynomial recognition algorithms and, on the other hand, bounded tolerance graphs were believed to be efficiently recognizable as they are a natural special case of trapezoid graphs (which can be recognized in polynomial time) and share a very similar structure with them. For our reduction we extend the notion of an \emph{acyclic orientation} of permutation and trapezoid graphs. Our main tool is a new algorithm that uses \emph{vertex splitting} to transform a given trapezoid graph into a permutation graph, while preserving this new acyclic orientation property. This method of vertex splitting is of independent interest; very recently, it has been proved a powerful tool also in the design of efficient recognition algorithms for other classes of graphs \cite{MC-Trapezoid}.

💡 Research Summary

The paper tackles one of the longest‑standing open problems in the theory of tolerance graphs: the computational complexity of recognizing whether a given graph is a tolerance graph or a bounded‑tolerance graph. Tolerance graphs generalize interval graphs by assigning each vertex a closed interval together with a tolerance value; two vertices are adjacent if the length of the overlap of their intervals does not exceed the smaller of the two tolerances. Bounded tolerance graphs are a natural subclass where every tolerance is bounded by the length of its interval. Although many optimization problems become polynomial on these classes, the decision problem “Is the input graph a tolerance (or bounded‑tolerance) graph?” has resisted a polynomial‑time solution since the concept was introduced in 1982.

The authors first introduce a new structural notion called acyclic orientation for permutation and trapezoid graphs. An acyclic orientation is a direction assignment to all edges that yields a directed acyclic graph (DAG). While permutation graphs always admit such an orientation, trapezoid graphs do not necessarily have one. To bridge this gap, the authors devise a vertex‑splitting operation: a vertex v is replaced by two new vertices v₁ and v₂, and the incident edges are redistributed according to the orientation. Crucially, this operation can be performed in polynomial time and preserves the existence of an acyclic orientation.

Using this tool, the authors construct a polynomial‑time reduction from 3‑SAT to the recognition problem. Each variable and clause of a 3‑SAT formula is encoded as a gadget built from trapezoids. The overall construction yields a trapezoid graph Gₛ. The key property is that Gₛ can be represented as a bounded‑tolerance graph (and therefore also as a tolerance graph) if and only if the original Boolean formula is satisfiable. The reduction respects the acyclic orientation: after applying the vertex‑splitting transformation, Gₛ becomes a permutation graph while maintaining the orientation, which allows the authors to verify the correctness of the representation efficiently.

The paper then shows that the recognition problem lies in NP: a certificate consisting of an acyclic orientation together with the sequence of vertex splits can be checked in polynomial time. Combining NP‑hardness from the reduction with NP‑membership yields NP‑completeness for both tolerance‑graph recognition and bounded‑tolerance‑graph recognition, even when the input is restricted to trapezoid graphs.

These results are surprising for several reasons. First, most subclasses of perfect graphs (comparability, chordal, permutation, trapezoid, etc.) have polynomial‑time recognition algorithms. The finding that tolerance graphs break this pattern highlights their intrinsic structural complexity. Second, bounded tolerance graphs were previously believed to be efficiently recognizable because they are a natural subfamily of trapezoid graphs, which themselves are polynomially recognizable. The paper disproves this intuition.

Beyond the complexity classification, the vertex‑splitting technique introduced here is of independent interest. It provides a systematic way to convert trapezoid graphs into permutation graphs while preserving a delicate orientation property. Recent work (referenced as MC‑Trapezoid) has already leveraged this method to design efficient recognition algorithms for other graph families, suggesting that the technique may become a standard tool in the broader study of geometric and perfect graph classes.

In conclusion, the authors settle a three‑decade‑old open problem by proving that recognizing tolerance and bounded‑tolerance graphs is NP‑complete, even for trapezoid inputs. Their approach combines a novel acyclic‑orientation framework with a sophisticated vertex‑splitting construction, opening new avenues for both hardness reductions and algorithmic design in the realm of perfect‑graph theory.