The Private Neighbor Cube

Authors:
Michael Fellows;Gerd Fricke;Stephen Hedetniemi;David Jacobs
Affiliations:
-;-;-;-
Venue:
SIAM Journal on Discrete Mathematics
Year:
1994

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Abstract

Let $S$ be a set of vertices in a graph $G = (V, E)$. The authors state that a vertex u in S has a private neighbor (relative to $S$) if either $u$ is not adjacent to any vertex in $S$ or $u$ is adjacent to a vertex $w$ that is not adjacent to any other vertex in $S$. Based on the notion of private neighbors, a set of eight graph theoretic parameters can be defined whose inequality relationships can be described by a three-dimensional cube. Most of these parameters have already been studied independently. This paper unifies this study and, helps to form a cohesive theory of private neighbors in graphs. Theoretical and algorithmic properties of this private neighbor cube are investigated, and many open questions are raised.