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

Abstract

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. $1\le |N(v)\cap S|\le 2$. Moreover a set $S\subseteq V$ is a total $[1,2]$-set if for each vertex of $V$, it is the case that $1\le |N(v)\cap S|\le 2$. 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.