Values of domination numbers of the queen's graph.
Various Bounds for Liar’s Domination Number
Let G = (V,E) be a graph. A set S ⊆ V is a dominating set if Uv∈S N[v] = V , where N[v] is the closed neighborhood of v. Let L ⊆ V be a dominating set, and let v be a designated vertex in V (an intruder vertex). Each vertex in L ∩ N[v] can report that v is the location of the intruder, but (at most) one x ∈ L ∩ N[v] can report any w ∈ N[x] as the intruder location or x can indicate that there is no intruder in N[x]. A dominating set L is called a liar’s dominating set if every v ∈ V (G) can be correctly...
Vertices contained in all minimum paired-dominating sets of a tree
A set of vertices in a graph is called a paired-dominating set if it dominates and contains at least one perfect matching. We characterize the set of vertices of a tree that are contained in all minimum paired-dominating sets of the tree.
Vertices Contained In All Or In No Minimum Semitotal Dominating Set Of A Tree
Let G be a graph with no isolated vertex. In this paper, we study a parameter that is squeezed between arguably the two most important domination parameters; namely, the domination number, γ(G), and the total domination number, γt(G). A set S of vertices in a graph G is a semitotal dominating set of G if it is a dominating set of G and every vertex in S is within distance 2 of another vertex of S. The semitotal domination number, γt2(G), is the minimum cardinality of a semitotal dominating set of...
Vizing-like conjecture for the upper domination of Cartesian products of graphs -- the proof.