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(...

In a graph , a set is said to be total dominating if every is adjacent to some member of . When the graph represents a communication network, a total dominating set corresponds to a collection of servers having a certain desirable backup property, namely, that every server is adjacent to some other server. Selfstabilization, introduced by Dijkstra [1, 2], is the most inclusive approach to fault...

Let denote the Cartesian product of graphs and A total dominating set of with no isolated vertex is a set of vertices of such that every vertex is adjacent to a vertex in The total domination number of is the minimum cardinality of a total dominating set. In this paper, we give a new lower bound of total domination number of using parameters total domination, packing and -domination numbers of ...

A set S of vertices of a graph G is a total dominating set if every vertex of V (G) is adjacent to some vertex in S. We provide three equivalent conditions for a tree to have a unique minimum total dominating set and give a constructive characterization of such trees.