Linear-Time Algorithms for the Paired-Domination Problem in Interval Graphs and Circular-Arc Graphs

In a graph $G$, a vertex subset $S\subseteq V(G)$ is said to be a dominating set of $G$ if every vertex not in $S$ is adjacent to a vertex in $S$. A dominating set $S$ of a graph $G$ is called a paired-dominating set if the induced subgraph $G[S]$ contains a perfect matching. The paired-domination problem involves finding a smallest paired-dominating set of $G$. Given an intersection model of an interval graph $G$ with sorted endpoints, Cheng et al. designed an $O(m+n)$-time algorithm for interval graphs and an $O(m(m+n))$-time algorithm for circular-arc graphs. In this paper, to solve the paired-domination problem in interval graphs, we propose an $O(n)$-time algorithm that searches for a minimum paired-dominating set of $G$ incrementally in a greedy manner. Then, we extend the results to design an algorithm for circular-arc graphs that also runs in $O(n)$ time.

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