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

In a graph G, a vertex subset S ⊆ 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. [9] 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.