Geodesic Convexity in Graphs

Pelayo, Ignacio M.

doi:10.1007/978-1-4614-8699-2

Cited by 77 publications

(54 citation statements)

References 84 publications

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“…All these axioms hold in particular for the so-called interval-convexities, where an interval I : X × X → 2 X has the property that x, y ∈ I(x, y), and convex sets are defined as the sets S such that I(x, y) ∈ S for any x, y ∈ S, see Calder [7]. Intervals in graphs usually arise from some types of paths, like the shortest (also called geodesic) or the induced (also called monophonic) paths; see the recent monograph [26] on geodesic convexity in graphs. Some early influential papers in the area were published by Farber and Jamison [16,17], and Duchet [14].…”

Section: Introductionmentioning

confidence: 99%

Toll convexity

Alcón

Brešar

Gologranc

et al. 2015

European Journal of Combinatorics

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a b s t r a c tA walk W between two non-adjacent vertices in a graph G is called tolled if the first vertex of W is among vertices from W adjacent only to the second vertex of W , and the last vertex of W is among vertices from W adjacent only to the second-last vertex of W . In the resulting interval convexity, a set S ⊂ V (G) is toll convex if for any two non-adjacent vertices x, y ∈ S any vertex in a tolled walk between x and y is also in S. The main result of the paper is that a graph is a convex geometry (i.e. satisfies the Minkowski-Krein-Milman property stating that any convex subset is the convex hull of its extreme vertices) with respect to toll convexity if and only if it is an interval graph. Furthermore, some well-known types of invariants are studied with respect to toll convexity, and toll convex sets in three standard graph products are completely described.

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Section: Introductionmentioning

confidence: 99%

Toll convexity

Alcón

Brešar

Gologranc

et al. 2015

European Journal of Combinatorics

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“…A path is named chordless if it has no chords (Pelayo, 2013) and we note that a path π between x and y is chordless if and only if Π P xy = {π} and thus, because by (14) it holds that ω(π, Σ P P ·P ) = σ xy·P , then (13) takes the form,…”

Section: Interpretation Of Covariance Path Weightsmentioning

confidence: 99%

Path weights in concentration graphs

Roverato

Castelo

2020

Biometrika

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A graphical model provides a compact and efficient representation of the association structure of a multivariate distribution by means of a graph. Relevant features of the distribution are represented by vertices, edges and other higher-order graphical structures, such as cliques or paths. Typically, paths play a central role in these models because they determine the independence relationships among variables. However, while a theory of path coefficients is available in models for directed graphs, little has been investigated about the strength of the association represented by a path in an undirected graph. Essentially, it has been shown that the covariance between two variables can be decomposed into a sum of weights associated with each of the paths connecting the two variables in the corresponding concentration graph. In this context, we consider concentration graph models and provide an extensive analysis of the properties of path weights and their interpretation. More specifically, we give an interpretation of covariance weights through their factorisation into a partial covariance and an inflation factor. We then extend the covariance decomposition over the paths of an undirected graph to other measures of association, such as the marginal correlation coefficient and a quantity that we call the inflated correlation. The usefulness of these finding is illustrated with an application to the analysis of dietary intake networks.

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“…For instance, shortest paths yield geodesic intervals, induced paths yield monophonic intervals etc. Each type of an interval then gives rise to the corresponding convexity, see [7,16] for some basic types of intervals/convexities.…”

Section: Introductionmentioning

confidence: 99%

“…They investigated the toll number and the t-hull number of a graph, which were investigated in terms of the geodesic convexity about 30 years ago [11,15] and intensively studied after that, for instance in graph products [3,4,6], in terms of other types of convexities [10,17] and more. See [8,9,16] for further reading on this topic.…”

Section: Introductionmentioning

confidence: 99%

Toll number of the strong product of graphs

Gologranc

Repolusk

2019

Discrete Mathematics

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A tolled walk T between two non-adjacent vertices u and v in a graph G is a walk, in which u is adjacent only to the second vertex of T and v is adjacent only to the second-to-last vertex ofis the union of toll intervals between all pairs of vertices from S. The size of a smallest set S whose toll closure is the whole vertex set is called a toll number of a graph G, tn(G). This paper investigates the toll number of the strong product of graphs. First, a description of toll intervals between two vertices in the strong product graphs is given. Using this result we characterize graphs with tn(G ⊠ H) = 2 and graphs with tn(G ⊠ H) = 3, which are the only two possibilities. As an addition, for the t-hull number of G ⊠ H we show that th(G ⊠ H) = 2 for any non complete graphs G and H. As extreme vertices play an important role in different convexity types, we show that no vertex of the strong product graph of two non complete graphs is an extreme vertex with respect to the toll convexity.

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Geodesic Convexity in Graphs

Cited by 77 publications

References 84 publications

Toll convexity

Toll convexity

Path weights in concentration graphs

Toll number of the strong product of graphs

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