Kernels for (connected) Dominating Set on graphs with Excluded Topological subgraphs
Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Dimitrios M., Thilikos

TL;DR
This paper establishes the first linear kernels for the (Connected) Dominating Set problems on H-topological minor free graphs, extending kernelization results to broader graph classes.
Contribution
It introduces a novel kernelization algorithm for (Connected) Dominating Set on H-topological-minor-free graphs, using new protrusion concepts and structural theorems.
Findings
Linear kernels for Dominating Set on H-topological-minor-free graphs.
Extension of kernelization results to broader graph classes.
Novel reduction rules based on graph structure.
Abstract
We give the first linear kernels for the (Connected) Dominating Set problems on H-topological minor free graphs. We prove the existence of polynomial time algorithms that, for a given H-topological-minor-free graph G and a positive integer k, output an H-topological-minor-free graph G' on O(k) vertices such that G has a (connected) dominating set of size k iff G' has one. Our results extend the known classes of graphs on which the Dominating Set and Connected Dominating Set problems admit linear kernels. Prior to our work, it was known that these problems admit linear kernels on graphs excluding a fixed apex graph H as a minor. Moreover, for Dominating Set, a kernel of size kc(H), where c(H) is a constant depending on the size of H, follows from a more general result on the kernelization of Dominating Set on graphs of bounded degeneracy. Alon and Gutner explicitly asked whether one can…
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Taxonomy
TopicsAdvanced Graph Theory Research · Graph Labeling and Dimension Problems · Complexity and Algorithms in Graphs
