Linear kernels for (connected) dominating set on H-minor-free graphs

We give the first linear kernels for DOMINATING SET and CONNECTED DOMINATING SET problems on graphs excluding a fixed graph H as a minor. In other words, we give polynomial time algorithms that, for a given H-minor free graph G and positive integer k, output an H-minor free graph G on O(k) vertices such that G has a (connected) dominating set of size k if and only if G has. Prior to our work, the only polynomial kernel for DOMINATING SET on graphs excluding a fixed graph H as a minor was due to Alon and Gutner [ECCC 2008, IWPEC 2009] and to Philip, Raman, and Sikdar [ESA 2009] but the size of their kernel is kc(H) , where c(H) is a constant depending on the size of H. Alon and Gutner asked explicitly, whether one can obtain a linear kernel for DOMINATING SET on H-minor free graphs. We answer this question in affirmative. For CONNECTED DOMINATING SET no polynomial kernel on H-minor free graphs was known prior to our work. Our results are based on a novel generic reduction rule prod...

• Type: Journal
• Year: 2012
• Where: SODA
• Authors: Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, Dimitrios M. Thilikos
• Keywords: Algorithms, Free Graphs, Monte Carlo Algorithm, Polynomial Time Algorithms, SODA 2012