Recognizing clique graphs of directed and rooted path graphs

AbstractWe describe characterizations for the classes of clique graphs of directed and rooted path graphs. The characterizations relate these classes to those of clique-Helly and strongly chordal graphs, respectively, which properly contain them. The characterizations lead to polynomial time algorithms for recognizing graphs of these classes

Efficient and Perfect domination on circular-arc graphs

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