Please use this identifier to cite or link to this item: https://hdl.handle.net/11147/13402
Title: Parity of an odd dominating set
Authors: Batal, Ahmet
Keywords: Lights out
All-ones problem
Odd dominating set
Parity domination
Domination number
Issue Date: 2022
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 $uin V(G)$. 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.
URI: https://doi.org/10.31801/cfsuasmas.1051208
https://search.trdizin.gov.tr/yayin/detay/1146262
https://hdl.handle.net/11147/13402
ISSN: 1303-5991
2618-6470
Appears in Collections:Mathematics / Matematik
TR Dizin İndeksli Yayınlar / TR Dizin Indexed Publications Collection

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