Efficient Algorithms for Circular-Arc Containment Graphs

Abstract

In the recent past, a wide variety of algorithms have been developed for a class of intersection graphs, called interval graphs. As a generalization of interval graphs, circular-arc graphs have also been studied extensively. Another category of graphs, namely containment graphs, has recently received some attention. In particular, interval containment graphs have been studied recently and several optimal algorithms have been developed for this class of graphs. In this thesis we introduce a new class of containment graphs called circular-arc containment graphs. A circular-arc containment graph is a generalization of an interval containment graph and is defined as follows: A graph G sub A = (V sub A, E sub A) is called a ciruclar-arc containment graph for a family A = {A sub 1, A sub n} of arcs on a circle, if for each v sub i V sub A, there is an arc A sub i A, and (v sub i, v sub j) E sub A if and only if one of A sub i and A sub j contains the other. We characterize this class of graphs by establishing its equivalence to another relatively new class of intersection graphs, called circular permutation graphs. Given a circular-arc containment graph in the form of a family of arcs on a circle, we develop efficient algorithms for finding a maximum clique, a maximum independent set, and a minimum coloring of the graph.