For a simple graph $G$ with vertex set $V(G)=\{v_1,...,v_n\}$, we define the closed neighborhood of $u$ as \\$N[u]=\{v \in V(G) \; | v \text{is adjacent to} u \text{or} v=u \}$ and matrix $N(G)$ whose $i$th column is characteristic vector $N[v_i]$. We say $S$ odd dominating if $N[u]\cap S$ for all $u\in V(G)$. prove that parity cardinality an equal to rank $G$, where defined dimension space $N(...

Given a graph G = (V, E) of order n and an n-dimensional non-negative vector d = (d(1), d(2), . . . , d(n)), called demand vector, the vector domination (resp., total vector domination) is the problem of finding a minimum S ⊆ V such that every vertex v in V \S (resp., in V ) has at least d(v) neighbors in S. The (total) vector domination is a generalization of many dominating set type problems,...

Let k be a positive integer and let G = (V,E) be a simple graph. The k-tuple domination number γ×k(G) of G is the minimum cardinality of a k-tuple dominating set S, a set that for every vertex v ∈ V , |NG[v]∩S| ≥ k. Also the total k-domination number γ×k,t(G) of G is the minimum cardinality of a total k -dominating set S, a set that for every vertex v ∈ V , |NG(v)∩S| ≥ k. The k-transversal numb...