# Algorithmic aspects of total Roman $\{2\}$-domination in graphs

Document Type : Original paper

Authors

1 National Institute of Technology, Warangal

2 National Institute of Technology Warangal

Abstract

For a simple, undirected, connected graph $G$, a function $h : V \rightarrow \lbrace 0, 1, 2 \rbrace$ is called a total Roman $\{2\}$-dominating function (TR2DF) if for every vertex $v$ in $V$ with weight $0$, either there exists a vertex $u$ in $N_G(v)$ with weight $2$, or at least two vertices $x, y$ in $N_G(v)$ each with weight $1$, and the subgraph induced by the vertices with weight more than zero has no isolated vertices. The weight of TR2DF $h$ is $\sum_{p \in V} h(p)$. The problem of determining  TR2DF of minimum weight is called minimum total Roman \{2\}-domination problem (MTR2DP). We show that MTR2DP is polynomial time solvable for bounded treewidth graphs, threshold graphs and chain graphs. We design a $2 (\ln(\Delta - 0.5) + 1.5)$-approximation algorithm for the MTR2DP and show that the same cannot have $(1 - \delta) \ln |V|$ ratio approximation algorithm  for any $\delta > 0$ unless $P = NP$.  Next, we show that MTR2DP is APX-hard for graphs with $\Delta=4$.  Finally, we show that the domination and TR2DF problems are not equivalent in computational complexity aspects.

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