On the computer enumeration of finite topologies

a b s t r a c tA subset S of vertices of a graph G is g-convex if whenever u and v belong to S, all vertices on shortest paths between u and v also lie in S. The g-spectrum of a graph is the set of sizes of its g-convex sets. In this paper we consider two problems -counting g-convex sets in a graph, and determining when a graph has g-convex sets of every cardinality (such graphs are said to have the continuum property). We show that the problem of counting g-convex sets of a graph whose components have diameter at most 2 is #P-complete, but for the class of cographs these sets can be enumerated in linear time. The problem of determining whether or not the g-convexity of a graph has the continuum property is proven to be NP-complete. While every graph is shown to be an induced subgraph of a graph whose gconvexity possesses the continuum property, graphs with the continuum property are rare since for any fixed ϵ ∈ (0, 1) it is shown that almost all n-vertex graphs have a gap in their g-spectrum of size at least Ω(n 1−ϵ ). Moreover, it is shown that for almost all graphs, every g-convex set is a clique, from which it follows that the number of g-convex sets in a random graph is at least n c ln n for some constant c. The graph convexity under discussion fits within the class of alignments on a finite set, namely those set systems on a finite set V that contain the whole set, the empty set, and are closed under intersection. Finite topologies are perhaps the most famous examples of alignments, and our results here are compared and contrasted with what can be said for topologies on a finite set.

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“…, n}. There is a well-known correspondence between topologies on [n] and preorders on [n], see [18]. , is a basis for the topology T .…”

Section: Resultsmentioning

confidence: 99%

On the spectrum and number of convex sets in graphs

Brown

Oellermann

2015

Discrete Mathematics

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“…We use |X| for the cardinality of a set X. Ever since the pioneer work of Evans, Harary and Lynn [2], counting such topologies can be done by counting digraphs as follows. For a preorder R on X, let D(R) be the direct graph whose vertex set is X and arc set is R \ ∆(X) = {(x, y) : x ̸ = y and (x, y) ∈ R}.…”

Section: Introductionmentioning

confidence: 99%

Enumerations of Finite Topologies Associated with a Finite Simple Graph

Kim¹,

Kwon²,

Lee³

2014

Kyungpook mathematical journal

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Abstract. The number of topologies (non-homeomorphic topologies) on a fixed finite set having n elements are now known up to n = 18 (n = 16 respectively) but still no complete formula yet. There are one to one correspondences among topologies, preorders and transitive digraphs on a given finite set. In this article, we enumerate topologies and non-homeomorphic topologies whose underlying graph is a given finite simple graph.

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“…[13][14][15]) between topologies on X and preorders on X is a bijection. Partial orders on X correspond to T 0 topologies on X.…”

Section: Introductionmentioning

confidence: 99%

Bounding the Roots of Ideal and Open Set Polynomials

et al. 2005

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Let P be a preorder (i.e., reflexive, transitive relation) on a finite set X. The ideal polynomial of P is the function ideal P (x) = ∑ k≥0 d k x k , where d k is the number of ideals (i.e. downwards closed sets) of cardinality k in P. We provide upper bounds for the moduli of the roots of ideal P (x) in terms of the width of P. We also provide examples of preorders with roots of large moduli. The results have direct applications to the generating polynomials counting open sets in finite topologies.

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On the computer enumeration of finite topologies

Cited by 67 publications

References 2 publications

On the spectrum and number of convex sets in graphs

On the spectrum and number of convex sets in graphs

Enumerations of Finite Topologies Associated with a Finite Simple Graph

Bounding the Roots of Ideal and Open Set Polynomials

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