abolfazl poureidi

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 $(|V|^{O(\log\log|V |)})$.

Let $G=(V,E)$ be a graph.  A double Roman dominating function  (DRDF) of   $G $   is a function   $f:Vto {0,1,2,3}$  such that, for each $vin V$ with $f(v)=0$,  there is a vertex $u $  adjacent to $v$  with $f(u)=3$ or there are vertices $x$ and $y $  adjacent to $v$  such that  $f(x)=f(y)=2$ and for each $vin V$ with $f(v)=1$,  there is a vertex $u $    adjacent to $v$    with  $f(u)>1$.  The weight of a DRDF $f$ is   $ f (V) =sum_{ vin V} f (v)$.   Let $n$ and  $k$ be integers such that  $3leq 2k+ 1  leq n$.  The   generalized Petersen graph $GP (n, k)=(V,E) $  is the  graph  with  $V={u_1, u_2,ldots,  u_n}cup{v_1, v_2,ldots, v_n}$ and $E={u_iu_{i+1}, u_iv_i, v_iv_{i+k}:  1 leq i leq n}$, where  addition is taken  modulo $n$. In this paper,  we firstly   prove that the  decision     problem  associated with   double Roman domination is NP-omplete even restricted to planar bipartite graphs with maximum degree at most 4.  Next, we   give  a dynamic programming algorithm for  computing a minimum DRDF (i.e., a  DRDF   with minimum weight  along  all   DRDFs)  of $GP(n,k )$  in $O(n81^k)$ time and space  and so a  minimum DRDF  of $GP(n,O(1))$  can be computed in $O( n)$ time and space.

Let $G=(V,E)$ be a given graph of order $n $. A function $f : V to {0,1, 2}$ is an independent Roman dominating function (IRDF) on $G$ if for every vertex $vin V$ with $f(v)=0$ there is a vertex $u$ adjacent to $v$ with $f(u)=2$ and ${vin V:f(v)> 0}$ is an independent set. The weight of an IRDF $f$ on $G $ is the value $f(V)=sum_{vin V}f(v)$. The minimum weight of an IRDF among all IRDFs on $G$ is called the independent Roman domination number of~$G$. In this paper, we give algorithms for computing the independent Roman domination number of $G$ in $O(|V|)$ time when $G=(V,E)$ is a tree, unicyclic graph or proper interval graph.

Let $n$ and $k$ be integers such that $3leq 2k+ 1 leq n$.The generalized Petersen graph $GP(n, k)=(V,E) $ is the graph with $V={u_1, u_2,ldots, u_n}cup{v_1, v_2,ldots, v_n}$ and $E={u_iu_{i+1}, u_iv_i, v_iv_{i+k}: 1 leq i leq n}$, whereaddition is in modulo $n$. A subset $Dsubseteq V$ is a dominating set of $GP(n, k)$ if for each $vin Vsetminus D$ there is a vertex $uin D$ adjacent to $v$. The minimum cardinality of a dominating set of $GP(n, k)$ is called the domination number of $GP(n, k)$.
In this paper we give a dynamic programming algorithm for computing the domination number of a given $GP(n,k )$ in $mathcal{O}(n)$ time and space for every $k=mathcal{O}(1)$.

Let $G=(V,E)$ be a graph. A doubleRoman dominating function (DRDF) on $G$ is a function$f:Vto{0,1,2,3}$ such that for every vertex $vin V$if $f(v)=0$, then either there is a vertex $u$ adjacent to $v$ with $f(u)=3$ orthere are vertices $x$ and $y$ adjacent to $v$ with $f(x)=f(y)=2$ and if $f(v)=1$, then there is a vertex $u$ adjacent to $v$ with$f(u)geq2$.A DRDF $f$ on $G$ is a total DRDF (TDRDF) if for any $vin V$ with $f(v)>0$ there is a vertex $u$ adjacent to $v$ with $f(u)>0$.The weight of $f$ is the sum $f(V)=sum_{vin V}f(v)$. The minimum weight of a TDRDF on $G$ is the total double Romandomination number of $G$. In this paper, we give a linear algorithm to compute thetotal double Roman domination number of agiven tree.

‎Let $G=(V,E)$ be a graph‎. ‎A subset $Ssubset V$ is a hop dominating set‎‎if every vertex outside $S$ is at distance two from a vertex of‎‎$S$‎. ‎A hop dominating set $S$ which induces a connected subgraph‎ ‎is called a connected hop dominating set of $G$‎. ‎The‎‎connected hop domination number of $G$‎, ‎$ gamma_{ch}(G)$,‎‎‎ ‎is the minimum cardinality of a connected hop‎‎dominating set of $G$‎. ‎A hop‎‎Roman dominating function (HRDF) of a graph $G$ is a function $‎‎f‎: ‎V(G)longrightarrow {0‎, ‎1‎, ‎2} $ having the property that‎‎for every vertex $ v in V $ with $ f(v) = 0 $ there is a‎‎vertex $ u $ with $ f(u)=2 $ and $ d(u,v)=2 $‎.‎The weight of‎‎an HRDF $ f $ is the sum $f(V) = sum_{vin V} f(v) $‎. ‎The‎‎minimum weight of an HRDF on $ G $ is called the hop Roman‎‎domination number of $ G $ and is denoted by $ gamma_{hR}(G)‎‎$‎. ‎We give an algorithm‎‎that decides whether $gamma_{hR}(T)=2gamma_{ch}(T)$ for a given‎‎tree $T$.‎‎{bf Keywords:} hop dominating set‎, ‎connected hop dominating set‎, ‎hop Roman dominating‎‎function‎.