Abstract
We consider the following problem: can a certain graph parameter of some given graph be reduced by at least \(d\) for some integer \(d\) via at most \(k\) edge contractions for some given integer \(k\)? We examine three graph parameters: the chromatic number, clique number and independence number. For each of these graph parameters we show that, when \(d\) is part of the input, this problem is polynomial-time solvable on \(P_4\)-free graphs and NP-complete as well as W[1]-hard, with parameter \(d\), for split graphs. As split graphs form a subclass of \(P_5\)-free graphs, both results together give a complete complexity classification for \(P_\ell \)-free graphs. The W[1]-hardness result implies that it is unlikely that the problem is fixed-parameter tractable for split graphs with parameter \(d\). But we do show, on the positive side, that the problem is polynomial-time solvable, for each parameter, on split graphs if \(d\) is fixed, i.e., not part of the input. We also initiate a study into other subclasses of perfect graphs, namely cobipartite graphs and interval graphs.
Öznur Yaşar Diner—Supported partially by Marie Curie International Reintegration Grant PIRG07/GA/2010/268322.
Daniël Paulusma—Supported by EPSRC EP/K025090/1.
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Diner, Ö.Y., Paulusma, D., Picouleau, C., Ries, B. (2015). Contraction Blockers for Graphs with Forbidden Induced Paths. In: Paschos, V., Widmayer, P. (eds) Algorithms and Complexity. CIAC 2015. Lecture Notes in Computer Science(), vol 9079. Springer, Cham. https://doi.org/10.1007/978-3-319-18173-8_14
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