Maximum weight independent sets and cliques in intersection graphs of filaments

Abstract

We describe a method of defining new families of graphs called G -mixed, using the way of partitioning the edge set of overlap graphs. Consider a hereditary family G of graphs. An oriented graph G(V,E) is called G -mixed if its edge set can be partitioned into two disjoint subsets E 1,E 2 such that G(V,E 1)∈ G, G(V,E 2) is transitive and for every three vertices w,v,u if w→v∈E 2 and (u,v)∈E 1 then (u,w)∈E 1 ; the letter G is generic and is replaced by specific names. The G -mixed graphs have a polynomial time algorithm to find maximum weight cliques, when G has such an algorithm. We define a new family of intersection graphs called interval-filament graphs which contain the polygon-circle graphs, the circle graphs, the chordal graphs and the cocomparability graphs. Let I be a family of intervals on a line L. In the plane, above L, construct to each interval i∈I a curve f i connecting its two endpoints, such that if two intervals are disjoint, their curves do not intersect; FI={f i∣i∈I} is a family of interval filaments and its intersection graph is an interval-filament graph. We prove that a graph is an interval-filament graph iff its complement is a cointerval-mixed graph. Since cointerval graphs have a polynomial time algorithm to find maximum weight cliques, we can find maximum weight independent sets in interval-filament graphs using the algorithm for maximum weight cliques in cointerval-mixed graphs. Interval-filament graphs have also an algorithm to find maximum weight cliques. New families of intersection graphs of filaments are defined using families of circular-arcs of a circle and families of subtrees of a tree or of a cactus.