THE RECOGNITION OF SIMPLE-TRIANGLE GRAPHS AND OF LINEAR-INTERVAL ORDERS IS POLYNOMIAL

Intersection graphs of geometric objects have been extensively studied, due to both 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 simple-triangle graphs. Simple-triangle graphs-also known as PI (point-interval) graphs-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. Given a graph G with n vertices, such that its complement (G) over bar has m edges, our algorithm runs in O(n(2)m) time. As a consequence, our algorithm also solves a longstanding open problem in the area of partial orders, namely, the recognition of linear-interval orders, i.e., of partial orders P = P-1 boolean AND 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 P-1 and P-2. In complete contrast to this, partial orders P which are the intersection of orders from the same class P have been extensively investigated, and in most cases the complexity status of these recognition problems has been already established.