Polynomial kernels for dominating set in graphs of bounded degeneracy and beyond

We show that for every fixed <i>j</i> ≥ <i>i</i> ≥ 1, the <i>k</i>-D<scp>ominating</scp> S<scp>et</scp> problem restricted to graphs that do not have <i>K<sub>ij</sub></i> (the complete bipartite graph on (<i>i</i> + <i>j</i>) vertices, where the two parts have <i>i</i> and <i>j</i> vertices, respectively) as a subgraph is fixed parameter tractable (FPT) and has a polynomial kernel. We describe a polynomial-time algorithm that, given a <i>K<sub>i,j</sub></i>-free graph <i>G</i> and a nonnegative integer <i>k</i>, constructs a graph <i>H</i> (the “kernel”) and an integer <i>k</i>' such that (1) <i>G</i> has a dominating set of size at most <i>k</i> if and only if <i>H</i> has a dominating set of size at most <i>k</i>', (2) <i>H</i> has <i>O</i>((<i>j</i> + 1)<sup><i>i</i> + 1</sup> <i>k<sup>i</sup><sup>2</sup></i>) vertices, and (3) <i>k</i>' = <i>O</i>((<i>j</i> + 1)<sup><i>i</i> + 1</sup> <i>k<sup>i</sup><sup>2</sup></i>). Since <i>d</i>-degenerate graphs do not have <i>K<sub>d+1,d+1</sub></i> as a subgraph, this immediately yields a polynomial kernel on <i>O</i>((<i>d</i> + 2)<sup><i>d</i>+2</sup> <i>k</i><sup>(<i>d</i> + 1)</sup><sup>2</sup>) vertices for the <i>k</i>-D<scp>ominating</scp> S<scp>et</scp> problem on <i>d</i>-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 <i>k</i>-D<scp>ominating</scp> S<scp>et</scp> is the class of <i>K<sub>h</sub></i>-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. <i>K<sub>h</sub></i>-topological-minor-free graphs are <i>K<sub>i,j</sub></i>-free for suitable values of <i>i,j</i> (but not vice-versa), and so our results show that <i>k</i>-D<scp>ominating</scp> S<scp>et</scp> has both FPT algorithms and polynomial kernels in strictly more general classes of graphs. Using the same techniques, we also obtain an <i>O</i>(<i>jk<sup>i</sup></i>) vertex-kernel for the <i>k</i>-I<scp>ndependent</scp> D<scp>ominating</scp> S<scp>et</scp> problem on <i>K<sub>i,j</sub></i>-free graphs.

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