Efficient and Perfect domination on circular-arc graphs

Given a graph $G=(V,E)$, a \emph{perfect dominating set} is a subset of vertices $V^{\mathrm{\prime }}\subseteq V(G)$ such that each vertex $v\in V(G)\setminus V^{\mathrm{\prime }}$ is dominated by exactly one vertex $v^{\mathrm{\prime }}\in V^{\mathrm{\prime }}$. An \emph{efficient dominating set} is a perfect dominating set $V^{\mathrm{\prime }}$ where $V^{\mathrm{\prime }}$ 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.