In this paper we consider the (d, n)-domination number, γd,n(Qn), the distance-d domination number γd(Qn) and the connected distance-d domination number γc,d(Qn) of ndimensional hypercube graphs Qn. We show that for 2 ≤ d ≤ bn/2c, and n ≥ 4, γd,n(Qn) ≤ 2n−2d+2, improving the bound of Xie and Xu [19]. We also show that γd(Qn) ≤ 2n−2d+2−r, for 2 − 1 ≤ n − 2d + 1 < 2 − 1, and γc,d(Qn) ≤ 2n−d, for ...

Let $D$ be a finite and simple digraph with vertex set $V(D)$‎.‎A signed total Roman $k$-dominating function (STR$k$DF) on‎‎$D$ is a function $f:V(D)rightarrow{-1‎, ‎1‎, ‎2}$ satisfying the conditions‎‎that (i) $sum_{xin N^{-}(v)}f(x)ge k$ for each‎‎$vin V(D)$‎, ‎where $N^{-}(v)$ consists of all vertices of $D$ from‎‎which arcs go into $v$‎, ‎and (ii) every vertex $u$ for which‎‎$f(u)=-1$ has a...

the inflation $g_{i}$ of a graph $g$ with $n(g)$ vertices and $m(g)$ edges is obtained from $g$ by replacing every vertex of degree $d$ of $g$ by a clique, which is isomorph to the complete graph $k_{d}$, and each edge $(x_{i},x_{j})$ of $g$ is replaced by an edge $(u,v)$ in such a way that $uin x_{i}$, $vin x_{j}$, and two different edges of $g$ are replaced by non-adjacent edges of $g_{i}$. t...

let $d$ be a finite and simple digraph with vertex set $v(d)$‎.‎a signed total roman $k$-dominating function (str$k$df) on‎‎$d$ is a function $f:v(d)rightarrow{-1‎, ‎1‎, ‎2}$ satisfying the conditions‎‎that (i) $sum_{xin n^{-}(v)}f(x)ge k$ for each‎‎$vin v(d)$‎, ‎where $n^{-}(v)$ consists of all vertices of $d$ from‎‎which arcs go into $v$‎, ‎and (ii) every vertex $u$ for which‎‎$f(u)=-1$ has a...

We limit our discussion to graphs that are simple and finite of order . Although 8 we often identify a graph with its set of vertices, in cases where we need to be K explicit we write . A set of vertices of is said to Z ÐKÑ Q K dominate K provided each vertex of is either in or adjacent to a vertex of . K Q Q The domination number of is the minimum order of a dominating set. A K dominating prov...

A subset D of vertices of a graph G is a dominating set if for each u ∈ V (G) \ D, u is adjacent to somevertex v ∈ D. The domination number, γ(G) ofG, is the minimum cardinality of a dominating set of G. A setD ⊆ V (G) is a total dominating set if for eachu ∈ V (G), u is adjacent to some vertex v ∈ D. Thetotal domination number, γt (G) of G, is theminimum cardinality of a total dominating set o...