k - Interval - filament graphs

For a fixed k, an oriented graph is k-transitive if it is acyclic and for every directed path p=u1→u2→...→uk+2 with k+2 vertices, p induces a clique if each of the two subpaths u1→u2→...→uk+1 and u2→...→uk+2 induces a clique. We describe an algorithm to find a maximum weight clique in a k-transitive graph. Consider a hereditary family G of graphs. A graph H(V,E) is called G-k-mixed if its edge set can be partitioned into two disjoint subsets E1, E2-E3 such that H(V,E1)∈G, H(V,E2) is transitive, H(V,E2-E3) is k-transitive and for every three distinct vertices u,v,w if u→v∈E2 and (v,w)∈E1 then (u,w)∈E1. The letter G is generic and can be replaced by names of specific families. We show that if the family G has a polynomial time algorithm to find a maximum clique, then, under certain restrictions there exists a polynomial time algorithm to find a maximum clique in a family of G-k-mixed graphs. Let I be a family of intervals on a line L in a plane PL such that every two intersecting intervals have a common segment. In PL, above L, construct to each interval i(v)∈I a filament v connecting its two endpoints, such that for every two filaments u,v having u∩v≠φ and disjoint intervals i(u)<i(v), there are no k filaments w with i(u)<i(w)<i(v) which intersect neither u nor v, are mutually disjoint and have mutually disjoint intervals. This is a family of k-interval-filaments and its intersection graph is a kinterval-filament graph; their complements are (k-transitive) mixed graphs and have a polynomial time algorithm for maximum cliques. Now, when two filaments u,v do not intersect because u⊂v, and between u and v there are at most k-1 non-intersecting filaments w1,...,wk-1 such that wi⊂wi+1 and intersect neither u nor v, we attach to each one of u,v a branch in the space above PL such that the two branches intersect. This is a family of general-k-interval-filaments and its intersection graph is a general-k-interval-filament graph; their complements are (k-transitive)-k-mixed graphs.