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 NPhard 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) · n algorithm on perfect graphs on n vertices where f is some (exponential) 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). We obtain the following results: 1. Vertex Partization on perfect graphs is FPT when parameterized by k + r + l. 2. The problem does not admit any polynomial sized kernel when parameterized by k + r + l. In other words, in polynomial time, the input graph can not 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.