Geevarghese Philip
Venkatesh Raman
Abstract
We show that for every fixed j ≥ i ≥ 1, the k-Dominating Set problem restricted to graphs that do not have Kij (the complete bipartite graph on (i + j) vertices, where the two parts have i and j vertices, respectively) as a subgraph is fixed parameter tractable (FPT) and has a polynomial kernel. We describe a polynomial-time algorithm that, given a Ki,j-free graph G and a nonnegative integer k, constructs a graph H (the “kernel”) and an integer k' such that (1) G has a dominating set of size at most k if and only if H has a dominating set of size at most k', (2) H has O((j + 1)i + 1 ki2) vertices, and (3) k' = O((j + 1)i + 1 ki2).
Since d-degenerate graphs do not have Kd+1,d+1 as a subgraph, this immediately yields a polynomial kernel on O((d + 2)d+2 k(d + 1)2) vertices for the k-Dominating Set problem on d-degenerate graphs, solving an open problem posed by Alon and Gutner [Alon and Gutner 2008; Gutner 2009].
The most general class of graphs for which a polynomial kernel was previously known for k-Dominating Set is the class of Kh-topological-minor-free graphs [Gutner 2009]. Graphs of bounded degeneracy are the most general class of graphs for which an FPT algorithm was previously known for this problem. Kh-topological-minor-free graphs are Ki,j-free for suitable values of i,j (but not vice-versa), and so our results show that k-Dominating Set has both FPT algorithms and polynomial kernels in strictly more general classes of graphs.
Using the same techniques, we also obtain an O(jki) vertex-kernel for the k-Independent Dominating Set problem on Ki,j-free graphs.
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Index Terms
- Polynomial kernels for dominating set in graphs of bounded degeneracy and beyond
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