A k-rainbow dominating function of a graph G is a map f from V (G) to the set of all subsets of {1, 2, . . . , k} such that {1, . . . , k} = ⋃ u∈N(v) f(u) whenever v is a vertex with f(v) = ∅. The k-rainbow domination number of G is the invariant γrk(G), which is the minimum sum (over all the vertices of G) of the cardinalities of the subsets assigned by a k-rainbow dominating function. We focu...

A graph is 2-stratified if its vertex set is partitioned into two classes, where the vertices in one class are colored red and those in the other class are colored blue. Let F be a 2-stratified graph rooted at some blue vertex v. An F -coloring of a graph G is a red-blue coloring of the vertices of G in which every blue vertex v belongs to a copy of F rooted at v. The F -domination number γF (G...

The weakly connected domination subdivision number sdγw(G) of a connected graph G is the minimum number of edges which must be subdivided (where each edge can be subdivided at most once) in order to increase the weakly connected domination number. The graph is strongγw-subdivisible if for each edge uv ∈ E(G) we have γw(Guv) > γw(G), where Guv is a graph G with subdivided edge uv. The graph is s...

During the last thirty years, the concept of domination in graphs has generated an impressive interest. A recent bibliography on the subject contains more than 1200 references and the number of new definitions is continually increasing. Rather than trying to give a catalogue of all of them, we survey the most classical and important notions (as independent domination, irredundant domination, k-...

for a graph $g$ let $gamma (g)$ be its domination number. we define a graph g to be (i) a hypo-efficient domination graph (or a hypo-$mathcal{ed}$ graph) if $g$ has no efficient dominating set (eds) but every graph formed by removing a single vertex from $g$ has at least one eds, and (ii) a hypo-unique domination graph (a hypo-$mathcal{ud}$ graph) if $g$ has at least two minimum dominating sets...

a set $s$ of vertices in a graph $g$ is a dominating set if every vertex of $v-s$ is adjacent to some vertex in $s$. the domination number $gamma(g)$ is the minimum cardinality of a dominating set in $g$. the annihilation number $a(g)$ is the largest integer $k$ such that the sum of the first $k$ terms of the non-decreasing degree sequence of $g$ is at most the number of edges in $g$. in this p...

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 set D of vertices in a connected graph G is called a k-dominating set if every vertex in G − D is within distance k from some vertex of D. The k-domination number of G, γk(G), is the minimum cardinality over all k-dominating sets of G. A graph G is k-distance domination-critical if γk(G − x) < γk(G) for any vertex x in G. This work considers properties of k-distance domination-critical graphs...

The k-domination number of a graph is the minimum size of a set X such that every vertex of G is in distance at most k from X. We give a linear time constant-factor approximation algorithm for k-domination number in classes of graphs with bounded expansion, which include e.g. proper minor-closed graph classes, classes closed on topological minors or classes of graphs that can be drawn on a fixe...