Parameterized algorithms on perfect graphs for deletion to (r,ℓ)-Graphs

For fixed integers r, ℓ ≥ 0, a graph G is called an (r,ℓ)-graph if the vertex set V (G) can be partitioned into r independent sets and ℓ cliques. Such a graph is also said to have cochromatic number r + ℓ. The class of (r, ℓ) graphs generalizes r-colourable graphs (when ℓ = 0) and hence not surprisingly, determining whether a given graph is an (r, ℓ)-graph is NP-hard even when r ≥ 3 or ℓ ≥ 3 in general graphs. When r and ℓ are part of the input, then the recognition problem is NP-hard even if the input graph is a perfect graph (where the Chromatic Number problem is solvable in polynomial time). It is also known to be fixed-parameter tractable (FPT) on perfect graphs when parameterized by r and ℓ. I.e. There is an f(r+ℓ) ·nO(1) algorithm on perfect graphs on n vertices where f is a function of r and ℓ. Observe that such an algorithm is unlikely on general graphs as the problem is NP-hard even for constant r and ℓ. In this paper, we consider the parameterized complexity of the following problem, which we call Vertex Partization. Given a perfect graph G and positive integers r, ℓ, k decide whether there exists a set S ⊆ V (G) of size at most k such that the deletion of S from G results in an (r, ℓ)-graph. This problem generalizes well studied problems such as Vertex Cover (when r = 1 and ℓ = 0), Odd Cycle Transversal (when r = 2, ℓ= 0) and Split Vertex Deletion (when r = 1 = ℓ). 1. Vertex Partization on perfect graphs is FPT when parameterized by k + r + ℓ. 2. The problem, when parameterized by k + r + ℓ, does not admit any polynomial sized kernel, under standard complexity theoretic assumptions. In other words, in polynomial time, the input graph cannot be compressed to an equivalent instance of size polynomial in k + r + ℓ In fact, our result holds even when k = 0. 3. When r, ℓ are universal constants, then Vertex Partization on perfect graphs, parameterized by k, has a polynomial sized kernel.