‎A Roman dominating function (RDF) on a graph G=(V,E) is a function  f : V → {0, 1, 2}  such that every vertex u for which f(u)=0 is‎ ‎adjacent to at least one vertex v for which f(v)=2‎. ‎An RDF f is called‎‎an outer independent Roman dominating function (OIRDF) if the set of‎‎vertices assigned a 0 under f is an independent set‎. ‎The weight of an‎‎OIRDF is the sum of its function values over ...

The domination game is played on a graph G by two players, named Dominator and Staller. They alternatively select vertices of G such that each chosen vertex enlarges the set of vertices dominated before the move on it. Dominator’s goal is that the game is finished as soon as possible, while Staller wants the game to last as long as possible. It is assumed that both play optimally. Game 1 and Ga...

A set S of vertices in a graphG is a total dominating set, denoted by TDS, ofG if every vertex ofG is adjacent to some vertex in S (other than itself). The minimum cardinality of a TDS ofG is the total domination number ofG, denoted by t(G). IfG does not contain K1,3 as an induced subgraph, then G is said to be claw-free. It is shown in [D. Archdeacon, J. Ellis-Monaghan, D. Fischer, D. Froncek,...

A graph G is said to be k-γt-critical if the total domination number γt(G) = k and γt(G + uv) < k for every uv / ∈ E(G). A k-γc-critical graph G is a graph with the connected domination number γc(G) = k and γc(G + uv) < k for every uv / ∈ E(G). Further, a k-tvc graph is a graph with γt(G) = k and γt(G− v) < k for all v ∈ V (G), where v is not a support vertex (i.e. all neighbors of v have degre...

for any $k in mathbb{n}$, the $k$-subdivision of graph $g$ is a simple graph $g^{frac{1}{k}}$, which is constructed by replacing each edge of $g$ with a path of length $k$. in [moharram n. iradmusa, on colorings of graph fractional powers, discrete math., (310) 2010, no. 10-11, 1551-1556] the $m$th power of the $n$-subdivision of $g$ has been introduced as a fractional power of $g$, denoted by ...

Henning, A.M. and H.C. Swart, Bounds relating generalized domination parameters, Discrete Mathematics 120 (1993) 933105. The domination number r(G) and the total domination number y,(G) of a graph G are generalized to the K,-domination number 7x,(G) and the total K,-domination number y;_(G) for n>2, where y(G)=y,,(G) and Y~(G)=~~~(G), K,-connectivity is defined and, for every integer n >2, the ...