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Discrete convexity: Retractions, morphisms and the partition problem. (English) Zbl 0957.05030

Balakrishnan, R. (ed.) et al., Graph connections. Proceedings of the conference, Cochin, India, January 28-31, 1998. New Delhi: Allied Publishers Limited. 10-18 (1999).
Summary: Introducing certain types of morphisms for general (abstract) convexity spaces, we give several ways for reducing the (Calder-)Eckhoff partition problem to simpler equivalent forms (finite, point-convex, interval spaces; restricted form, i.e. with distinct points). With additional results (to appear in a forthcoming paper) we show how the general problem can be reduced to its restricted version for finite graph-geodetic convexities. For instance, for every convexity space with Helly number \(h\) and partition numbers \(p_k\), we provide (constructively) a finite connected graph whose geodetic convexity has Helly number \(h\), Radon number \(p_2\) and \(k\)th partition number not less than \(p_k\).
For the entire collection see [Zbl 0947.00020].

MSC:

05C12 Distance in graphs
52A01 Axiomatic and generalized convexity
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
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