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Organizers |
Every empire has a defendable border region
by
Paul C. Kainen
Georgetown University
The title is a colloquial description of the following Lemma: For every connected plane graph G and every cycle z of G, either z is itself the boundary of a region or else there is a region r of the graph such that r is contained inside z and the boundary of r intersects z in a set homeomorphic to the closed unit interval I. This implies that the Mac Lane cycle basis of a plane graph is "robust" - i.e., if z is a cycle, the basis cycles, of which z is the sum, can be ordered such that each successive cycle after the first intersects the mod-2 sum of the previous terms in a homeomorph of I. Hence, all of the partial sums remain cycles. Implications for the algebra of commutative diagrams will be briefly discussed.
Date received: November 13, 2015
Copyright © 2015 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cblw-24.