A Linear-Time Algorithm for the Maximum Matched-Paired-Domination Problem in Cographs

Let $G=(V,E)$ be a graph without isolated vertices. A matching in $G$ is a set of independent edges in $G$. A perfect matching $M$ in $G$ is a matching such that every vertex of $G$ is incident to an edge of $M$. A set $S\subseteq V$isa\mathit{\text{paired-dominating}}\mathit{\text{set}}of$G$ifeveryvertexin$V-S$ is adjacent to some vertex in $S$ and if the subgraph $G[S]$ induced by $S$ contains at least one perfect matching. The paired-domination problem is to find a paired-dominating set of $G$ with minimum cardinality. A set $MPD\subseteq E$ is a $\mathit{\text{matched-paired-dominating}}\mathit{\text{set}}$ of $G$ if $MPD$ is a perfect matching of $G[S]$ induced by a paired-dominating set $S$ of $G$. Note that the paired-domination problem can be regard as finding a matched-paired-dominating set of $G$ with minimum cardinality. Let $\mathcal{R}$ be a subset of $V$, $MPD$ be a matched-paired-dominating set of $G$, and let $V(MPD)$ denote the set of vertices being incident to edges of $MPD$. A $\mathit{\text{maximum}}\mathit{\text{matched-paired-dominating}}\mathit{\text{set}}$ $MMPD$ of $G$ w.r.t. $\mathcal{R}$ is a matched-paired-dominating set such that $|V(MMPD)\cap $\mathcal{R}$|\geqslant |V(MPD)\cap $\mathcal{R}$|$\mathrm{.}Anedgein$MPD$ is called $\mathit{\text{free-paired-edge}}$ if neither of its both vertices is in $\mathcal{R}$. Given a graph $G$ and a subset $\mathcal{R}$ of vertices of $G$, the $\mathit{\text{maximum}}\mathit{\text{matched-paired-domination}}\mathit{\text{problem}}$ is to find a maximum matched-paired-dominating set of $G$ with the least free-paired-edges; note that, if $\mathcal{R}$ is empty, the stated problem coincides with the classical paired-domination problem. In this paper, we present a linear-time algorithm to solve the maximum matched-paired-domination problem in cographs.