Complexity of Paired Domination Problems on Circle and $k$-Polygon Graphs

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.