# Barriers in metric spaces

Dress, A., Moulton, Vincent, Spillner, Andreas and Wu, Taoyang (2009) Barriers in metric spaces. Applied Mathematics Letters, 22 (8). pp. 1150-1153.

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## Abstract

Defining a subset B of a connected topological space T to be a barrier (in T) if B is connected and its complement T-B is disconnected, we will investigate barriers B in the tight span View the MathML source Turn MathJax on of a metric D defined on a finite set X (endowed, as a subspace of RX, with the metric and the topology induced by the l8-norm) that are of the form B=Be(f)?{g?T(D):?f-g?8=e} Turn MathJax on for some f?T(D) and some e=0. In particular, we will present some conditions on f and e which ensure that such a subset of T(D) is a barrier in T(D). More specifically, we will show that Be(f) is a barrier in T(D) if there exists a bipartition (or split) of the e-support View the MathML source of f into two non-empty sets A and B such that f(a)+f(b)=ab+e holds for all elements a?A and b?B while, conversely, whenever Be(f) is a barrier in T(D), there exists a bipartition of View the MathML source into two non-empty sets A and B such that, at least, f(a)+f(b)=ab+2e holds for all elements a?A and b?B.

Item Type: Article metric space,tight span,cutpoint Faculty of Science > School of Computing Sciences http://dx.doi.org/10.1016/j.aml.2008.10.... Vishal Gautam 11 Mar 2011 16:22 21 Apr 2020 17:35 https://ueaeprints.uea.ac.uk/id/eprint/22263 10.1016/j.aml.2008.10.006