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 (Inf Process Lett 24:133–137, 1987) showed that this problem is NP-complete even on split graphs if q is part of input, but gave an $$n^{O(q)}$$ algorithm on chordal graphs. We first observe that the problem is W[2]-hard when parameterized by q, even on split graphs. However, when parameterized by $$\ell $$ , 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. Finally, we show that (under standard complexity-theoretic assumption) the problem does not admit a polynomial kernel on split and perfect graphs in the following sense:
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