余剰1体での被覆：トポロジーを見る
Covering with Excess One: Seeing the Topology
グリッド領域上の被覆空間のトポロジーを研究し、最小数N+1体の被覆エージェントによる空間が領域形状によらず1次元複体のホモトピー型を持つことを示し、オイラー標数の公式を提示した。
著者: Han Wang
分類: math.GN, cs.CG, cs.RO
原文アブストラクト
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.
