Parameterized algorithms 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 ' cliques. This brings us to the following natural parameterized questions: Vertex (r, ℓ)-Partization and Edge (r, ℓ)-Partization. An input to these problems consist of a graph G and a positive integer k and the objective is to decide whether there exists a set S ⊆ V (G) (S ⊆ E(G)) such that the deletion of S from G results in an (r, ℓ)-graph. These problems generalize well studied problems such as Odd Cycle Transversal, Edge Odd Cycle Transversal, Split Vertex Deletion and Split Edge Deletion. We do not hope to get parameterized algorithms for either Vertex (r, ℓ)-Partization or Edge (r, ℓ)- Partization when either of r or ℓ is at least 3 as the recognition problem itself is NP-complete. This leaves the case of r, ℓ ∈ {1, 2}. We almost complete the parameterized complexity dichotomy for these problems by obtaining the following results: 1. We show that Vertex (r, ℓ)-Partization is fixed parameter tractable (FPT) for r, ℓ ∈ {1, 2}. Then we design an O(√log n)-factor approximation algorithms for these problems. These approximation algorithms are then utilized to design polynomial sized randomized Turing kernels for these problems. 2. Edge (r, ℓ)-Partization is FPT when (r, ℓ) ∈ {(1, 2), (2, 1)}. However, the parameterized complexity of Edge (2, 2)-Partization remains open. For our approximation algorithms and thus for Turing kernels we use an interesting finite forbidden induced graph characterization, for a class of graphs known as (r, ℓ)-split graphs, properly containing the class of (r, ℓ)-graphs. This approach to obtain approximation algorithms could be of an independent interest.