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(...

A set S of vertices of a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. The independent domination number of G, denoted by i(G), is the minimum cardinality of an independent dominating set of G. In this paper, some new classes of graphs with equal domination and independent domination numbers are presented and exa...