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Hooshmandasl, P. Sharifani","submitted_at":"2017-07-20T11:02:48Z","abstract_excerpt":"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. 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