Steiner distance and convexity in graphs

Cáceres, José; Márquez, Alberto; Puertas, María Luz

doi:10.1016/j.ejc.2007.03.007

Cited by 44 publications

(31 citation statements)

References 11 publications

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“…In a similar way, totally balanced hypergraphs and strongly chordal graphs have been characterized as convex geometries of some particular (hyper)graph convexities [16]. More recently, the so-called Steiner convexity was introduced, and it was shown that precisely 3-fan-free chordal graphs are convex geometries with respect to this convexity [6].…”

Section: Introductionmentioning

confidence: 97%

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: 97%

Toll convexity

Alcón

Brešar

Gologranc

et al. 2015

European Journal of Combinatorics

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“…Further results involving Steiner tree problems related to geodesic convexity in graphs can be found in [15,18,26,28,47,55,96,115,116,136,150,152,155,172,180].…”

Section: Theorem 512 ([119]) Every Edge Steiner Set Of a Connected mentioning

confidence: 99%

Geodesic Convexity in Graphs

Pelayo¹

2013

SpringerBriefs in Mathematics

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SpringerBriefs in Mathematics showcases expositions in all areas of mathematics and applied mathematics. Manuscripts presenting new results or a single new result in a classical field, new field, or an emerging topic, applications, or bridges between new results and already published works, are encouraged. The series is intended for mathematicians and applied School of Agricultural Engineering of Barcelona

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“…While several convexity notions exist for graphs (e.g., g-convexity [7], m-convexity [5,7], ap-convexity [4], tpconvexity [3], Steiner convexity [2,10]), fewer convexity notions have been defined explicitly for hypergraphs. The first hypergraph convexity that has been introduced is simple-path convexity (sp-convexity, for short) [7], which is a generalization of ap-convexity.…”

Section: Introductionmentioning

confidence: 99%