A Polynomial time Algorithm for the Maximum Weight Independent Set Problem on Outerstring Graphs∗

Outerstring graphs are the intersection graphs of curves that lie inside a disk such that each curve intersects the boundary of the disk in one of its endpoints. Outerstring graphs were introduced in 1991 and are amongst the most general classes of intersection graphs studied, including among others, chordal graphs and interval filament graphs. To date no polynomial time algorithm is known for any of the classical graph optimization problems on outerstring graphs, in fact most are NP-hard. It is known that there is an intersection model for any outerstring graph that consists of polygonal arcs attached to a circle. However, this representation may require an exponential number of segments relative to the size of the graph. We develop a general simple dynamic programming algorithm for the Maximum Weight Independent Set problem. Given an outerstring graph and an intersection model consisting of polygonal arcs with a total of N segments, we show that our algorithm solves the Maximum Weight Independent Set problem in O ( N ) time. If the polygonal arcs are restricted to single segments, then outersegment graphs result. In this special case we show that our algorithm solves the Maximum Weight Independent Set problem in O ( n ) time where n is the number of vertices in the graph. ∗Supported by NSERC †Corresponding Author