apx-complete

Given a graph $G=(V,E)$,  a  function  $f:V\to \{0,1,2,3,4\}$ is a triple Roman  dominating function (TRDF)  of $G$, for each vertex $v\in V$,  (i) if $f (v ) = 0 $, then  $v$ must have either one neighbour in $V_4$, or either two neighbours in $V_2 \cup  V_3$ (one neighbour in $V_3$) or either three neighbours in $V_2 $, (ii) if $f (v ) = 1 $, then $v$ must have either one neighbour in  $V_3 \cup  V_4$  or either two neighbours in $V_2 $, and if $f (v ) = 2 $, then $v$ must have one neighbour in $V_2 \cup  V_3\cup  V_4$. The triple Roman  domination number of $G$ is the  minimum weight of an TRDF  $f$  of $G$, where the weight of $f$ is $\sum_{v\in V}f(v)$.  The triple  Roman  domination problem is to compute the  triple Roman  domination number of a given graph.  In this paper, we study the triple  Roman  domination problem. We show that   the problem is NP-complete for  the  star convex bipartite  and the   comb convex bipartite graphs and is APX-complete for graphs of degree at~most~4. We propose a linear-time algorithm for computing  the triple Roman  domination number of proper interval graphs.  We also   give an $( 2 H(\Delta(G)+1) -1  )$-approximation algorithm  for solving the problem  for any graph $G$,  where   $  \Delta(G)$ is the maximum degree of $G$ and $H(d)$ denotes the first $d$ terms of the harmonic  series. In addition, we prove  that  for any $\varepsilon>0$  there is no  $(1/4-\varepsilon)\ln|V|$-approximation  polynomial-time   algorithm for solving  the problem on bipartite and split  graphs, unless NP $\subseteq$ DTIME $(|V|^{O(\log\log|V |)})$.

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.

A set $S subseteq V(G)$ is a semitotal dominating set of a graph $G$ if it is a dominating set of $G$ and every vertex in $S$ is within distance 2 of another vertex of $S$. The semitotal domination number $gamma_{t2}(G)$ is the minimum cardinality of a semitotal dominating set of $G$. We show that the semitotal domination problem is APX-complete for bounded-degree graphs, and the semitotal domination problem in any graph of maximum degree $Delta$ can be approximated with an approximation ratio of $2+ln(Delta-1)$.