Steiner intervals, geodesic intervals, and betweenness

Brešar, Boštjan; Changat, Manoj; Mathews, J. Howard; Peterin, Iztok; Narasimha-Shenoi, Prasanth G.; Horvat, Aleksandra Tepeh

doi:10.1016/j.disc.2009.05.022

Cited by 16 publications

(9 citation statements)

References 23 publications

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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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“…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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“…Since then many papers have appeared on the topic. For a small collection see [1,4,9,13,14,20,22,25,28] and the references therein. In particular, note that Steiner number was introduced by Chartrand and Zhang in [14].…”

Section: Introductionmentioning

confidence: 99%

“…In [4,9] the authors worked on a variation of the Steiner problem, namely for multi sets (instead of sets). These definitions coincide for Steiner convexity, since we take all subsets of C in the definition of a Steiner convex set C and not only k-subsets of C for a fixed k. (The latter is known as the k-Steiner convexity.)…”

Section: Introductionmentioning

confidence: 99%

Some Steiner concepts on lexicographic products of graphs

Anand

Changat

Peterin

et al. 2014

Discrete Math. Algorithm. Appl.

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Let G be a graph and W a subset of V(G). A subtree with the minimum number of edges that contains all vertices of W is a Steiner tree for W. The number of edges of such a tree is the Steiner distance of W and union of all vertices belonging to Steiner trees for W form a Steiner interval. We describe both of these for the lexicographic product of graphs. We also give a complete answer for the following invariants with respect to the Steiner convexity: the Steiner number, the rank, the hull number, and the Carathéodory number, and a partial answer for the Radon number.

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“…. , u n ) and its betweenness properties has been initiated recently in [1]. There specifically, the class of graphs for which the Steiner interval S(u 1 , u 2 , .…”

Section: Introductionmentioning

confidence: 99%

“…, u n ) = ∪ i =j I(u i , u j ) is characterized. This property is defined as the the union property of the n-Steiner interval in [1]. When n = 2 the union property trivially holds in all graphs.…”

Section: Introductionmentioning

confidence: 99%

n-ary transit functions in graphs

Changat¹,

Mathews²,

Narasimha-Shenoi³

et al. 2010

Discuss. Math. Graph Theory

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n-ary transit functions are introduced as a generalization of binary (2-ary) transit functions. We show that they can be associated with convexities in natural way and discuss the Steiner convexity as a natural n-ary generalization of geodesicaly convexity. Furthermore, we generalize the betweenness axioms to n-ary transit functions and discuss the connectivity conditions for underlying hypergraph. Also n-ary all paths transit function is considered.

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Steiner intervals, geodesic intervals, and betweenness

Cited by 16 publications

References 23 publications

Geodesic Convexity in Graphs

Geodesic Convexity in Graphs

Some Steiner concepts on lexicographic products of graphs

n-ary transit functions in graphs

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