凸体的切交图分类
Classifying Convex Bodies by their Contact and Intersection Graphs
摘要：
假设$A$是平面中的凸体，并且$A_1,\dots,A_n$是$A$的平移。这样的平移产生$A$、$G=(V,E)$的交叉图，其具有顶点$V=\{1,\dots,n\}$和边$E=\{uv\mid A_u\cap A_v\neq \emptyset\}$。满足$E'\subset E$是$A_u$和$A_v$内部不相交的边集$uv$的子图$G'=(V, E')$是$A$的单位距离图.此外，如果$G'=G$，即，如果每当$u\neq v$时，$A_u$和$A_v$的内部是不相交的，则$G$是$A$的接触图。在本文中，我们研究哪些凸体对具有相同的接触，单位距离，或相交图。我们说两个凸体$A$和$B$是等价的，如果存在$B$的线性变换$B'$，使得对于任何斜率，分别包含在$A$和$B'$中的具有该斜率的最长线段是等长的。对于广泛的一类凸体，包括所有严格凸体和正多边形的线性变换，我们证明了$A$和$B$的接触图是相同的当且仅当$A$和$B$是等价的。我们证明了相同的声明为单位距离和相交图。
英文摘要：
Suppose that $A$ is a convex body in the plane and that $A_1,\dots,A_n$ are translates of $A$. Such translates give rise to an intersection graph of $A$, $G=(V,E)$, with vertices $V=\{1,\dots,n\}$ and edges $E=\{uv\mid A_u\cap A_v\neq \emptyset\}$. The subgraph $G'=(V, E')$ satisfying that $E'\subset E$ is the set of edges $uv$ for which the interiors of $A_u$ and $A_v$ are disjoint is a unit distance graph of $A$. If furthermore $G'=G$, i.e., if the interiors of $A_u$ and $A_v$ are disjoint whenever $u\neq v$, then $G$ is a contact graph of $A$. In this paper we study which pairs of convex bodies have the same contact, unit distance, or intersection graphs. We say that two convex bodies $A$ and $B$ are equivalent if there exists a linear transformation $B'$ of $B$ such that for any slope, the longest line segments with that slope contained in $A$ and $B'$, respectively, are equally long. For a broad class of convex bodies, including all strictly convex bodies and linear transformations of regular polygons, we show that the contact graphs of $A$ and $B$ are the same if and only if $A$ and $B$ are equivalent. We prove the same statement for unit distance and intersection graphs.
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