Complexity of Paired Domination Problems on Circle and $k$-Polygon Graphs
Ta-Yu Mu, Ching-Chi Lin
TL;DR
This work studies paired-domination on circle and $k$-polygon graphs. It first establishes NP-completeness for finding a minimum paired-dominating set on circle graphs via a Hamiltonian-path reduction, introducing a circle-graph construction with a targeted PD-size of $2n^2+2n-2$. For the broader class of $k$-polygon graphs, the authors develop two polynomial-time algorithms: a minimum dominating set algorithm running in $O\left(n\left(\frac{n}{k^2-k}\right)^{2k^2-4k}\right)$ and a minimum paired-dominating set algorithm running in $O\left(n\left(\frac{n}{k^2-k}\right)^{2k^2-2k}\right)$, leveraging the outer boundary $O$ and inner boundary $I$ and the permutation-graph structure of subproblems. They further improve the domination algorithm to $O(n^{3k-5})$ time and extend it to total domination with the same bound, using boundary-based decomposition and the blossom algorithm for matching. Collectively, these results advance the understanding of domination variants on circle and $k$-polygon graphs and suggest avenues for approximation and tighter time bounds in related graph classes.
Abstract
A set $D \subseteq V$ is a dominating set of a graph $G$ if every vertex in $V - D$ is adjacent to at least one vertex in $D$. A dominating set $D$ is a paired-dominating set if the subgraph of $G$ induced by $D$ contains a perfect matching. In this paper, we prove that determining the minimum paired-dominating set in circle graphs is NP-complete. We further present an $O(n(\frac{n}{k^2-k})^{2k^2-2k})$-time algorithm for finding the minimum paired-dominating set in $k$-polygon graphs, a subclass of circle graphs. Additionally, we refine the existing algorithm of Elmallah and Stewart for computing the minimum dominating set in $k$-polygon graphs, reducing its time complexity from $O(n^{4k^2+3})$ to $O(n^{3k-5})$, and further extend it to find the minimum total dominating set.