Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Parity of the cardinality of an odd dominating set   For a simple graph $G$ with vertex set $V(G)=\{v_1,...,v_n\}$, we define the
closed neighborhood set of a vertex $u$ as $N[u]= \{v \in V(G) \; | \; v \;
\text{is adjacent to} \; u \; \text{or} \; v=u \}$ and the closed neighborhood
matrix $N(G)$ as the matrix whose $i$th column is the characteristic vector of
$N[v_i]$. We say a set $S$ is odd dominating if $N[u]\cap S$ is odd for all
$u\in V(G)$. We prove that the parity of the cardinality of an odd dominating
set of $G$ is equal to the parity of the rank of $G$, where rank of $G$ is
defined as the dimension of the column space of $N(G)$. Using this result we
prove several corollaries in one of which we obtain a general formula for the
nullity of the join of graphs.

1
The categories of this paper are: cs
4
The categories of this paper are: cond-mat
36
The categories of this paper are: cmp-lg
19
The categories of this paper are: eess
28
The categories of this paper are: alg-geom
24
The categories of this paper are: dg-ga
27
The categories of this paper are: comp-gas
12
The categories of this paper are: nucl-th
29
The categories of this paper are: funct-an
17
The categories of this paper are: q-fin
32
The categories of this paper are: atom-ph
31
The categories of this paper are: ao-sci
22
The categories of this paper are: chao-dyn
37
The categories of this paper are: supr-con
10
The categories of this paper are: q-bio
6
The categories of this paper are: astro-ph