A Linear Algorithm for Computing $γ_{_{[1,2]}}$-set in Generalized Series-Parallel Graphs - Pouyeh Sharifani; Mohammad Reza Hooshmandasl (Transactions on Combinatorics, 2020)
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. $1 ≤ \vert N(v) \cap S \vert ≤ 2$. Moreover a set $S \subseteq V$ is a total $[1,2]$-set if for each vertex of $V$, it is the case that $1 ≤ \vert N(v) \cap S \vert ≤ 2$. The $[1,2]$-domination number of $G$, denoted $γ_{[1,2]}(G)$, is the minimum number of vertices in a $[1,2]$-set. Every $[1,2]$-set with cardinality of $γ_{[1,2]}(G)$ is called a $γ_{[1,2]}$-set. Total $[1,2]$-domination number and $γ_{t[1,2]}$-sets of $G$ are defined in a similar way. This paper presents a linear time algorithm to find a $γ_{[1,2]}$-set and a $γ_{t[1,2]}$-set in generalized series-parallel graphs.