Parameterized Algorithms for Max Colorable Induced Subgraph Problem on Perfect Graphs

Abstract

We address the parameterized complexity of Max Colorable Induced Subgraph on perfect graphs. The problem asks for a maximum sized q-colorable induced subgraph of an input graph G. Yannakakis and Gavril [IPL 1987] showed that this problem is NP-complete even on split graphs if q is part of input, but gave a n O(q) algorithm on chordal graphs. We first observe that the problem is W[2]-hard parameterized by q, even on split graphs. However, when parameterized by ℓ, the number of vertices in the solution, we give two fixed-parameter tractable algorithms. The first algorithm is efficient when the input graph contains only polynomially many maximal independent sets; for example split graphs and co-chordal graphs. The running time of the second algorithm is FPT in ℓ alone (whenever T α is a polynomial in n), since q ≤ ℓ for all non-trivial situations. Finally, we show that (under standard complexity-theoretic assumptions) the problem does not admit a polynomial kernel on split and perfect graphs in the following sense: (a) On split graphs, we do not expect a polynomial kernel if q is a part of the input. (b) On perfect graphs, we do not expect a polynomial kernel even for fixed values of q ≥ 2.