A New Intersection Model for Multitolerance Graphs, Hierarchy, and Efficient Algorithms

Tolerance graphs model interval relations in such a way that intervals can tolerate a certain degree of overlap without being in conflict. This class of graphs has attracted many research efforts, mainly due to its interesting structure and its numerous applications, while a number of variations of this model has appeared. In particular, one of the most natural generalizations of tolerance graphs, namely multitolerance graphs, has been introduced in 1998 [25], where two tolerances are allowed for each interval. These two tolerances – one from the left and one from the right side of the interval – define an infinite number of the so called tolerance-intervals, which make the multitolerance model inconvenient to cope with. The main subclass of multitolerance graphs, namely bounded multitolerance graphs, coincide with the widely known class of trapezoid graphs that has been extensively studied. In this article we introduce the first non-trivial intersection model for general multitolerance graphs, given by objects in the three-dimensional space called trapezoepipeds, which unifies in a simple and intuitive way the trapezoid representation for bounded multitolerance graphs and the recently introduced parallelepiped representation for tolerance graphs [22]. Apart from being important on its own, this new intersection model proves to be a powerful tool for designing efficient algorithms. Given a multitolerance graph with n vertices and m edges, we present three new algorithms that compute a minimum coloring and a maximum clique in optimal O(n log n) time, and a maximum weight independent set in O(m+ n log n) time – this algorithm also improves the best known running time of O(n) for the same problem on tolerance graphs [22]. Moreover, we prove several structural results on the class of multitolerance graphs, complementing thus the hierarchy of perfect graphs given in [11]. The resulting hierarchy of classes of perfect graphs is complete, i.e. all inclusions are strict.