Engineering Dominating Patterns: A Fine-grained Case Study

The Dominating \(H\)-Pattern problem generalizes the classical \(k\)-Dominating Set problem: for a fixed pattern \(H\) and a given graph \(G\), the goal is to find an induced subgraph \(S\) of \(G\) such that (1) \(S\) is isomorphic to \(H\), and (2) \(S\) forms a dominating set in \(G\). Fine-grained complexity results show that on worstcase inputs, any significant improvement over the naive brute-force algorithm is unlikely, as this would refute the Strong Exponential Time Hypothesis. Nevertheless, a recent work by Dransfeld et al. (ESA 2025) reveals some significant improvement potential particularly in sparse graphs.