University of IsfahanTransactions on Combinatorics2251-86579120200301A Linear Algorithm for Computing $gamma_{_{[1,2]}}$-set in Generalized Series-Parallel Graphs1242418510.22108/toc.2019.105482.1509ENPouyehSharifaniDepartment of Computer Science, Yazd University, Yazd, IranMohammad RezaHooshmandaslDepartment of Computer Science, Yazd University, Yazd, Iran0000-0002-3834-3610Journal Article20170720For 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.https://toc.ui.ac.ir/article_24185_5d3db88464af44f66f4256faa00162d7.pdf