In this paper, we continue the study of the domination game in graphs introduced by Bre{v{s}}ar, Klav{v{z}}ar, and Rall. We study the paired-domination version of the domination game which adds a matching dimension to the game. This game is played on a graph $G$ by two players, named Dominator and Pairer. They alternately take turns choosing vertices of $G$ such that each vertex chosen by Domin...

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...

Let $kge 1$ be an integer, and let $G$ be a finite and simple graph with vertex set $V(G)$.A weak signed Roman $k$-dominating function (WSRkDF) on a graph $G$ is a function$f:V(G)rightarrow{-1,1,2}$ satisfying the conditions that $sum_{xin N[v]}f(x)ge k$ for eachvertex $vin V(G)$, where $N[v]$ is the closed neighborhood of $v$. The weight of a WSRkDF $f$ is$w(f)=sum_{vin V(G)}f(v)$. The weak si...

For any integer $kgeq 1$ and any graph $G=(V,E)$ with minimum degree at least $k-1$‎, ‎we define a‎ ‎function $f:Vrightarrow {0,1,2}$ as a Roman $k$-tuple dominating‎ ‎function on $G$ if for any vertex $v$ with $f(v)=0$ there exist at least‎ ‎$k$ and for any vertex $v$ with $f(v)neq 0$ at least $k-1$ vertices in its neighborhood with $f(w)=2$‎. ‎The minimum weight of a Roman $k$-tuple dominatin...

A graph G is dot-critical if contracting any edge decreases the domination number. It is totally dot-critical if identifying any two vertices decreases the domination number. We show that the totally dot-critical graphs essentially include the much-studied domination vertex-critical and edge-critical graphs as special cases. We investigate these properties, and provide a characterization of dot...

A graph with no isolated vertices is edge critical with respect to total restrained domination if for any non-edge e of G, the total restrained domination number of G+ e is less than the total restrained domination number of G. We call these graphs γtr-edge critical. In this paper, we characterize all γtr-edge critical unicyclic graphs.

A set D ⊆ V of a graph G = (V,E) is called an outer-connected dominating set of G if for all v ∈ V , |NG[v] ∩ D| ≥ 1, and the induced subgraph of G on V \D is connected. The Minimum Outer-connected Domination problem is to find an outer-connected dominating set of minimum cardinality of the input graph G. Given a positive integer k and a graph G = (V,E), the Outer-connected Domination Decision ...

Let k ≥ 1 be an integer, and let G be a finite and simple graph with vertex set V (G). A signed total Italian k-dominating function (STIkDF) on a graph G is a functionf : V (G) → {−1, 1, 2} satisfying the conditions that $sum_{xin N(v)}f(x)ge k$ for each vertex v ∈ V (G), where N(v) is the neighborhood of $v$, and each vertex u with f(u)=-1 is adjacent to a vertex v with f(v)=2 or to two vertic...

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...

an equitable domination has interesting application in the contextof social networks. in a network, nodes with nearly equal capacitymay interact with each other in a better way. in the societypersons with nearly equal status, tend to be friendly. in thispaper, we introduce new variant of equitable domination of agraph. basic properties and some interesting results have beenobtained.