Covering with Excess One: Seeing the Topology

Han Wang

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arXiv · 1312.7477

Covering with Excess One: Seeing the Topology

Han Wang

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Abstract

We have initiated the study of topology of the space of coverings on grid domains. The space has the following constraint: while all the covering agents can move freely (we allow overlapping) on the domain, their union must cover the whole domain. A minimal number $N$ of the covering agents is required for a successful covering of the domain. In this paper, we demonstrate beautiful topological structures of this space on grid domains in 2D with $N+1$ coverings, the topology of the space has the homotopy type of $1$ dimensional complex, regardless of the domain shape. We also present the Euler characteristic formula which connects the topology of the space with that of the domain itself.

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Han Wang. 2013-12-28. Covering with Excess One: Seeing the Topology. https://arxiv.org/abs/1312.7477

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