A Linear Algorithm for Computing $\gamma_{_{[1,2]}}$-set in Generalized Series-Parallel Graphs

For a graph $G=(V,E)$, a set $S \subseteq V$ is a $[1,2]$-set if it is a dominating set for $G$ and each vertex $v \in V \setminus S$ is dominated by at most two vertices of $S$, i.e. $1 \leq \vert N(v) \cap S \vert \leq 2$. Moreover a set $S \subseteq V$ is a total $[1,2]$-set if for each vertex of $V$, it is the case that $1 \leq \vert N(v) \cap S \vert \leq 2$. The $[1,2]$-domination number of $G$, denoted $\gamma_{[1,2]}(G)$, is the minimum number of vertices in a $[1,2]$-set. Every $[1,2]$-set with cardinality of $\gamma_{[1,2]}(G)$ is called a $\gamma_{[1,2]}$-set. Total $[1,2]$-domination number and $\gamma_{t[1,2]}$-sets of $G$ are defined in a similar way. This paper presents a linear time algorithm to find a $\gamma_{[1,2]}$-set and a $\gamma_{t[1,2]}$-set in generalized series-parallel graphs.