- Communications Faculty of Sciences University Ankara Series A1 Mathematics and Statistics
- Vol: 71 Issue: 4
- Parity of an odd dominating set
Parity of an odd dominating set
Authors : Ahmet Batal
Pages : 1023-1028
Doi:10.31801/cfsuasmas.1051208
View : 10 | Download : 5
Publication Date : 2022-12-30
Article Type : Research
Abstract :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.Keywords : Lights out, all-ones problem, odd dominating set, parity domination, domination number