On Zero Forcing Number of Graphs and Their Complements
Abstract
The \emph{zero forcing number}, $Z(G)$, of a graph $G$ is the minimum cardinality of a set $S$ of black vertices (whereas vertices in $V(G) \setminus S$ are colored white) such that $V(G)$ is turned black after finitely many applications of "the colorchange rule": a white vertex is converted to a black vertex if it is the only white neighbor of a black vertex. Zero forcing number was introduced and used to bound the minimum rank of graphs by the "AIM Minimum Rank  Special Graphs Work Group". It's known that $Z(G)\geq \delta(G)$, where $\delta(G)$ is the minimum degree of $G$. We show that $Z(G)\leq n3$ if a connected graph $G$ of order $n$ has a connected complement graph $\overline{G}$. Further, we characterize a tree or a unicyclic graph $G$ which satisfies either $Z(G)+Z(\overline{G})=\delta(G)+\delta(\overline{G})$ or $Z(G)+Z(\overline{G})=2(n3)$.
 Publication:

arXiv eprints
 Pub Date:
 February 2014
 arXiv:
 arXiv:1402.1962
 Bibcode:
 2014arXiv1402.1962E
 Keywords:

 Mathematics  Combinatorics;
 05C50;
 05C05;
 05C38;
 05D99
 EPrint:
 9 pages, 5 figures. arXiv admin note: text overlap with arXiv:1204.2238