Intersection graphs are very important in both theoretical as well as application point of view. Depending on the geometrical representation, different type of intersection graphs are defined. Among them interval, circular-arc, permutation, trapezoid, chordal, disk, circle graphs are more important. In this article, a brief introduction of each of these intersection graphs is given. Some basic ...

A general method for the computation of various parameters measuring the vulnerability of a graph is introduced. Four measures of vulnerability are considered, i.e., the toughness, scattering number, vertex integrity and the size of a minimum balanced separator. We show how to compute these parameters by polynomial time algorithms for various classes of intersection graphs like permutation grap...

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 grap...

We present O(n’R + n3R3) time algorithms to compute the treewidth, pathwidth, minimum fill-in and minimum interval graph completion of asteroidal triple-free graphs, where n is the number of vertices and R is the number of minimal separators of the input graph. This yields polynomial time algorithms for the four NP-complete graph problems on any subclass of the asteroidal triple-free graphs tha...

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 simple-triangle graphs. Simple-triangle graphs – also known as PI graphs (for ...

We show that if a tolerance graph is the complement of a comparability graph it is a trapezoid graph, i.e., the complement of an order of interval dimension at most 2. As consequences we are able to give obstructions for the class of bounded tolerance graphs and to give an example of a graph which is alternatingly orientable but not a tolerance graph. We also characterize the tolerance graphs a...