06064595

j

06064595 Isaak, Garth Johnson, Peter Petrie, Caleb Integer and fractional security in graphs. Discrete Appl. Math. 160, No. 13-14, 2060-2062 (2012). 2012 Elsevier Science B.V. (North-Holland), Amsterdam EN secure sets Hall's theorem integer attack integer defense

doi:10.1016/j.dam.2012.04.018

Summary: Let $G=(V,E)$ be a graph. A subset $S$ of $V$ is said to be secure if it can defend itself from an attack by vertices in $N[S] - S$. In the usual definition, each vertex in $N[S] - S$ can attack exactly one vertex in $S$, and each vertex in $S$ can defend itself or one of its neighbors in $S$. A defense of $S$ is successful if each vertex has as many defenders as attackers. We look at allowing an attacking vertex to divide its one unit of attack among multiple targets, or allowing a defending vertex to divide its one unit of defense among multiple allies. Three new definitions of security are given. It turns out that two of the new definitions are the same as the original.