Domination When the Stars Are Out

We algorithmize the recent structural characterization for claw-free graphs by Chudnovsky and Seymour. Building on this result, we show that Dominating Set on claw-free graphs is (i) fixed-parameter tractable and (ii) even possesses a polynomial kernel. To complement these results, we establish that Dominating Set is not fixed-parameter tractable on the slightly larger class of graphs that exclude K_{1,4} as an induced subgraph (K_{1,4}-free graphs). We show that our algorithmization can also be used to show that the related Connected Dominating Set problem is fixed-parameter tractable on claw-free graphs. To complement that result, we show that Connected Dominating Set has no polynomial kernel on claw-free graphs and is not fixed-parameter tractable on K_{1,4}-free graphs. Combined, our results provide a dichotomy for Dominating Set and Connected Dominating Set on K_{1,L}-free graphs and show that the problem is fixed-parameter tractable if and only if L <= 3.

💡 Research Summary

The paper investigates the parameterized complexity of the Dominating Set (DS) and Connected Dominating Set (CDS) problems on claw‑free graphs and on the broader family of K₁,ℓ‑free graphs. The authors start from the deep structural theorem of Chudnovsky and Seymour, which classifies claw‑free graphs into three well‑understood categories: line graphs, proper circular‑arc (or “switch”) graphs, and graphs that can be decomposed into a small “core” together with pendant “leaf” structures. By turning this classification into an explicit polynomial‑time decomposition algorithm, the paper obtains a fixed‑parameter tractable (FPT) algorithm for DS on claw‑free graphs. The algorithm enumerates all possible selections of vertices from the core (bounded by the parameter k) and then greedily dominates the leaf parts, yielding a running time of f(k)·n^O(1).

Beyond mere tractability, the authors develop a polynomial‑size kernel for DS on claw‑free graphs. The kernelization rules identify groups of leaf vertices that share identical neighborhoods in the core; each such group can be replaced by a single representative without affecting the existence of a solution of size ≤ k. If any group exceeds k vertices, the instance is immediately a NO‑instance. Repeated application of these reductions shrinks the graph to O(k²) vertices, establishing a polynomial kernel.

The paper then turns to the slightly larger class of K₁,₄‑free graphs. Using a parameter‑preserving reduction from the Multi‑Colored Clique problem, the authors prove that DS remains W