Efficient and Perfect domination on circular-arc graphs

Given a graph G = (V,E), a perfect dominating set is a subset of vertices V ′ ⊆ V (G) such that each vertex v ∈ V (G) \ V ′ is dominated by exactly one vertex v ∈ V . An efficient dominating set is a perfect dominating set V ′ where V ′ is also an independent set. These problems are usually posed in terms of edges instead of vertices. Both problems, either for the vertex or edge variant, remains NP-Hard, even when restricted to certain graphs families. We study both variants of the problems for the circular-arc graphs, and show efficient algorithms for all of them.