Algorithms for vertex subset and vertex partitioning problems with runtime single exponential in rankwidth

Let σ and ρ be finite or co-finite subsets of natural numbers. A subset of vertices S of a graph G = (V,E) is a (σ, ρ) -set of G if ∀v ∈ V : |N(v) ∩ S| ∈  σ if v ∈ S ρ if v ∈ V \ S A degree constraint matrix Dq is a q by q matrix with entries being finite or co-finite subsets of natural numbers. A Dq -partition in a graph G = (V,E) is a partition V1, V2, ..., Vq of V such that for 1 ≤ i, j ≤ q we have ∀v ∈ Vi : |NG(v) ∩ Vj | ∈ Dq[i, j] For graphs of rankwidth k , given with a rank-decomposition of width k , we show how to solve any minimization or maximization problem over (σ, ρ) -sets in time O(n ∗ (m+ 2 2d+2kdk)) , with d being the maximum member of σ and ρ (or of the complement set if it is co-finite). We also show how to decide if the graph contains a Dq -partition in time O(n ∗ (m + 2 2dq+kkd2)) , now with d taken over all Dq[i, j] .