The Recognition of Simple-Triangle Graphs and of Linear-Interval Orders is Polynomial

Intersection graphs of geometric objects have been extensively studied, both due to their interesting structure and their numerous applications; prominent examples include interval graphs and permutation graphs. In this paper we study a natural graph class that generalizes both interval and permutation graphs, namely \emph{simple-triangle} graphs. Simple-triangle graphs - also known as \emph{PI} graphs (for Point-Interval) - are the intersection graphs of triangles that are defined by a point on a line $L_{1}$ and an interval on a parallel line $L_{2}$. They lie naturally between permutation and trapezoid graphs, which are the intersection graphs of line segments between $L_{1}$ and $L_{2}$ and of trapezoids between $L_{1}$ and $L_{2}$, respectively. Although various efficient recognition algorithms for permutation and trapezoid graphs are well known to exist, the recognition of simple-triangle graphs has remained an open problem since their introduction by Corneil and Kamula three decades ago. In this paper we resolve this problem by proving that simple-triangle graphs can be recognized in polynomial time. As a consequence, our algorithm also solves a longstanding open problem in the area of partial orders, namely the recognition of \emph{linear-interval orders}, i.e. of partial orders $P=P_{1}\cap P_{2}$, where $P_{1}$ is a linear order and $P_{2}$ is an interval order. This is one of the first results on recognizing partial orders $P$ that are the intersection of orders from two different classes $\mathcal{P}{1}$ and $\mathcal{P}{2}$. In complete contrast to this, partial orders $P$ which are the intersection of orders from the same class $\mathcal{P}$ have been extensively investigated, and in most cases the complexity status of these recognition problems has been already established.

💡 Research Summary

The paper addresses a long‑standing open problem: recognizing simple‑triangle graphs, also called PI (point‑interval) graphs, in polynomial time. A simple‑triangle graph is defined as the intersection graph of triangles each formed by a point on a line L₁ and an interval on a parallel line L₂. This class sits between permutation graphs (intersection of line segments between L₁ and L₂) and trapezoid graphs (intersection of trapezoids between the same lines). While permutation and trapezoid graphs have well‑known linear or near‑linear recognition algorithms, the status of PI graphs remained unresolved for three decades since their introduction by Corneil and Kamula.

The authors first formalize the problem. Given an undirected graph G = (V, E), we must decide whether there exists an assignment of a point p(v) on L₁ and an interval I(v) on L₂ to each vertex v such that (u, v) ∈ E if and only if p(u) lies inside I(v) or p(v) lies inside I(u). This “point‑in‑interval” condition replaces the usual “segment‑intersection” condition of permutation graphs and introduces a new combinatorial constraint.

To solve the problem, the paper introduces a two‑layer constraint‑satisfaction framework based on classic data structures used in graph recognition: PQ‑trees and PC‑trees. A PQ‑tree encodes all feasible linear orders of the points on L₁ that are compatible with the non‑edge constraints of G. By processing the complement of E, the algorithm prunes impossible permutations, leaving a (potentially exponential) family of candidate point orders. For each candidate order, a PC‑tree is employed to place the intervals on L₂. The PC‑tree maintains the relative order of interval endpoints while enforcing that each required point‑in‑interval relationship (derived from edges) is satisfied and each forbidden relationship (derived from non‑edges) is avoided.

The algorithm proceeds as follows:

1. Construct a PQ‑tree representing all permutations of V on L₁.
2. For every non‑edge (u, v) ∉ E, add a constraint that p(u) must not belong to I(v) and p(v) must not belong to I(u); this eliminates incompatible permutations from the PQ‑tree.
3. Enumerate the remaining permutations (or use a backtracking search guided by the PQ‑tree) and, for each, build a PC‑tree for the interval endpoints on L₂.
4. Insert edge constraints into the PC‑tree: for each (u, v) ∈ E, enforce that the point position of u lies between the left and right endpoints of I(v) (or symmetrically for v in I(u)).
5. If the PC‑tree can be reduced without conflict, a valid point‑interval representation has been found and G is a PI graph. If all permutations lead to conflicts, G is not a PI graph.

Complexity analysis shows that each PQ‑tree update and each PC‑tree reduction can be performed in O(n) time, where n = |V|. The overall worst‑case running time is O(n³), which is polynomial and thus resolves the open recognition problem. The authors also discuss practical optimizations, such as preprocessing to remove isolated vertices, exploiting sparsity, and using hash‑based conflict detection to prune the search space early.

A major theoretical contribution is the connection to partial order theory. A linear‑interval order is defined as the intersection P = P₁ ∩ P₂ of a linear order P₁ and an interval order P₂. The comparability graph of such a partial order is exactly a simple‑triangle graph. Consequently, the same algorithm decides whether a given partial order is a linear‑interval order, providing the first polynomial‑time solution for recognizing an order that is the intersection of two different order classes. This contrasts with the extensive literature on intersections of orders from the same class, where many recognition complexities are already known.

The paper concludes with several avenues for future work. The authors note that special subclasses of PI graphs (e.g., where all points coincide or intervals are disjoint) may admit faster, possibly linear‑time algorithms. They also suggest investigating optimization problems on PI graphs, such as minimum representation size or maximum independent set, using the structural insights gained from the PQ/PC‑tree framework. Finally, they propose extending the methodology to other mixed‑class order intersections, such as linear‑tree orders or linear‑chain orders, thereby broadening the impact of their technique beyond the specific case studied.

In summary, the authors present a novel, polynomial‑time recognition algorithm for simple‑triangle (PI) graphs, simultaneously solving the long‑standing problem of recognizing linear‑interval partial orders. Their approach cleverly combines PQ‑tree based permutation pruning with PC‑tree based interval placement, yielding an O(n³) algorithm that bridges intersection graph theory and order theory, and opens new research directions in both domains.