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# Parameterized Algorithms for MAX COLORABLE INDUCED SUBGRAPH Problem on Perfect Graphs

Misra, Neeldhara and Panolan, Fahad and Rai, Ashutosh and Raman, Venkatesh and Saurabh, Saket (2013) Parameterized Algorithms for MAX COLORABLE INDUCED SUBGRAPH Problem on Perfect Graphs. In: 39th International Workshop on Graph-Theoretic Concepts in Computer Science (WG), JUN 19-21, 2013, Lubeck, GERMANY, pp. 370-381. PDF gra_con_com_sci_8165_370_2013.pdf - Published Version Restricted to Registered users only Download (421kB) | Request a copy
Official URL: http://dx.doi.org/ 10.1007/978-3-642-45043-3_32

## Abstract

We address the parameterized complexity ofMaxColorable 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 W2]-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 runs in time 5.44(l) (n+#alpha(G))(O(1)) where #alpha(G) is the number of maximal independent sets of the input graph. The second algorithm runs in time q(l+o()l())n(O(1))T(alpha) where T-alpha 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 running time of the second algorithm is FPT in l alone (whenever T-alpha is a polynomial in n), since q <= l for all non-trivial situations. Finally, we show that (under standard complexitytheoretic 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.

Item Type: Conference Proceedings Copy right for this article belongs to the SPRINGER-VERLAG BERLIN, HEIDELBERGER PLATZ 3, D-14197 BERLIN, GERMANY Division of Electrical Sciences > Computer Science & Automation Id for Latest eprints 28 Nov 2014 05:47 28 Nov 2014 05:47 http://eprints.iisc.ac.in/id/eprint/50349 View Item