a function $f:v(g)rightarrow {-1,0,1}$ is a {em minusdominating function} if for every vertex $vin v(g)$, $sum_{uinn[v]}f(u)ge 1$. a minus dominating function $f$ of $g$ is calleda {em global minus dominating function} if $f$ is also a minusdominating function of the complement $overline{g}$ of $g$. the{em global minus domination number} $gamma_{g}^-(g)$ of $g$ isdefined as $gamma_{g}^-(g)=min{...

the broadcast domination number of the cartesian product of two cycles is completely determined.

The bondage number b(G) of a nonempty graph G is the cardinality of a smallest edge set whose removal from G results in a graph with domination number greater than the domination number (G) ofG. Kang andYuan proved b(G) 8 for every connected planar graph G. Fischermann, Rautenbach and Volkmann obtained some further results for connected planar graphs. In this paper, we generalize their results ...

The domination number of a graph is the cardinality of a smallest subset of its vertex set with the property that each vertex of the graph is in the subset or adjacent to a vertex in the subset. This graph parameter has been studied extensively since its introduction during the early 1960s and finds application in the generic setting where the vertices of the graph denote physical entities that...

The rainbow game domination subdivision number of a graph G is defined by the following game. Two players D and A, D playing first, alternately mark or subdivide an edge of G which is not yet marked nor subdivided. The game ends when all the edges of G are marked or subdivided and results in a new graph G′. The purpose of D is to minimize the 2-rainbow dominating number γr2(G ′) of G′ while A t...

A Roman domination function on a graph G is a function r : V (G) → {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of a Roman function is the value r(V (G)) = ∑ u∈V (G) r(u). The Roman domination number γR(G) of G is the minimum weight of a Roman domination function on G . "Roman Criticality" has been ...

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...

A subset S ⊆ V (G) is a double dominating set of G if S dominates every vertex of G at least twice. The double domination number dd(G) is the minimum cardinality of a double dominating set of G. The double domination subdivision number sddd(G) is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the double domination n...

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. Burton and Sumner [Discrete Math. 306 (2006) 11–18] posed the problem: Is it true that for k 4, there exists a totally k-dot-critical graph with no critical vertices? In this paper, we show that this problem has a positive...

A graph G is 3-domination critical if its domination number γ is 3 and the addition of any edge decreases γ by 1. It was proved by Favaron et al. that α ≤ δ + 2 for any connected 3-domination critical graph. Denote by τ (G) the toughness of a graph G . Recently Chen et al. conjectured that a connected 3-domination critical graph G is Hamilton-connected if and only if τ (G) > 1 and showed the co...