Convex sets in graphs, II. Minimal path convexity
TLDR
Caratheodory, Helly and Radon type theorems are proved for M-convex sets and the Helly number equals the size of a maximum clique.About:
This article is published in Journal of Combinatorial Theory, Series B.The article was published on 1988-06-01 and is currently open access. It has received 139 citations till now. The article focuses on the topics: Helly's theorem & Chordal graph.read more
Citations
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On local convexity in graphs
TL;DR: This work investigates the classes of graphs which are characterized by certain local convexity conditions with respect to geodesic conveXity, in particular, those graphs in which balls around nodes are convex, and those graph in which neighborhoods of convex sets are conveX.
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Complexity results related to monophonic convexity
TL;DR: It is proved that the decision problems corresponding to the m-convexity and monophonic numbers are NP-complete, and the size of a minimum subset whose convex hull is equal to V(G) (m-hull number) is described.
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Convexities related to path properties on graphs
TL;DR: This work proposes a more general approach for 'path properties' in graphs, focusing on the behaviour of such convexities on the Cartesian product of graphs and on the classical convexity invariants, such as the Caratheodory, Helly and Radon numbers in relation with graph invariants.
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On triangle path convexity in graphs
Manoj Changat,Joseph Mathew +1 more
TL;DR: Convexity invariants like Caratheodory, Helly and Radon numbers are computed for triangle path convexity in graphs like Helly, Radon and Carathodory.
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The All-Paths Transit Function of a Graph
TL;DR: A transit function R on a set V is characterized by transit axioms as mentioned in this paper, where R(u, u, u) is a transit function satisfying the axiomatization of U(U, U) for all paths in a connected graph.
References
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Convexity in graphs and hypergraphs
M. Farber,Robert E. Jamison +1 more
TL;DR: In this paper, the Minkowski-Krein-Milman theorem, Caratheodory's theorem and Tietze's convexity theorem were studied in graphs and hypergraphs.
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Axiomatic convexity theory and relationships between the Carathéodory, Helly, and Radon numbers
TL;DR: In this article, it was shown that if a set X has Carathέodory number c and Helly number h then X has Radon number r ch+1.
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On Helly's Theorem and the Axioms of Convexity
TL;DR: In this article, it was shown that when C 1,..., C m are convex domains in an n-dimensional Euclidean space and every set of n+I of these domains have a common point, then there exists a point which is common to all the domains.