### Parameterized Algorithms for Max Colorable Induced Subgraph Problem on Perfect Graphs [chapter]

Neeldhara Misra, Fahad Panolan, Ashutosh Rai, Venkatesh Raman, Saket Saurabh
2013 Lecture Notes in Computer Science
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-hard parameterized by q, even on split graphs. However, when parameterized by , the number of vertices in the
more » ... lution, we give two fixed-parameter tractable algorithms. -The first algorithm runs in time 5.44 (n + #α(G)) O(1) where #α(G) is the number of maximal independent sets of the input graph. -The second algorithm runs in time q +o( ) n O(1) Tα where Tα is the time required to find a maximum independent set in any induced subgraph of G. 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 second algorithm is FPT in alone, since q ≤ 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.