Abstract
Graphs and Algorithms In this paper we discuss the bounds of and relations among various kinds of intersection numbers of graphs. Especially, we address extremal graphs with respect to the established bounds. The uniqueness of the minimum-size intersection representations for some graphs is also studied. In the course of this work, we introduce a superclass of chordal graphs, defined in terms of a generalization of simplicial vertex and perfect elimination ordering.
Highlights
We consider only finite undirected graphs without parallel edges or self-loops
We present our main result, which confirms that edge clique graph is a natural context in investigating intersection numbers
Proof: First we show that C∈CG w(C) is a feasible solution to the linear program Covering Problem II (CPII), i.e., for each e ∈ E(G), we have e∈E(C) w(C) ≥ 1
Summary
We consider only finite undirected graphs without parallel edges or self-loops. Let G be a graph. The intersection number of a graph G is the minimum size of a multifamily representation of G, denoted i(G). If f assigns distinct sets to different vertices of G, f is called a family representation of G. f is a Helly multifamily representation of G if f satisfies the Helly property [18], namely for any clique K of G, it holds v∈V (K) f (v) = ∅.
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