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arXiv · 1506.00255

Critical and Maximum Independent Sets of a Graph

Adi Jarden·Vadim E. Levit·Eugen Mandrescu
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Abstract

Let G be a simple graph with vertex set V(G). A subset S of V(G) is independent if no two vertices from S are adjacent. By Ind(G) we mean the family of all independent sets of G while core(G) and corona(G) denote the intersection and the union of all maximum independent sets, respectively. The number d(X)= |X|-|N(X)| is the difference of the set of vertices X, and an independent set A is critical if d(A)=max{d(I):I belongs to Ind(G)} (Zhang, 1990). Let ker(G) and diadem(G) be the intersection and union, respectively, of all critical independent sets of G (Levit and Mandrescu, 2012). In this paper, we present various connections between critical unions and intersections of maximum independent sets of a graph. These relations give birth to new characterizations of Koenig-Egervary graphs, some of them involving ker(G), core(G), corona(G), and diadem(G).

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Adi Jarden, Vadim E. Levit, Eugen Mandrescu. 2015-05-31. Critical and Maximum Independent Sets of a Graph. https://arxiv.org/abs/1506.00255

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