嵌入树、外平面图和多足图的单位圆盘表示
Unit Disk Representations of Embedded Trees, Outerplanar and Multi-Legged Graphs
摘要：
图$G$的单位圆盘相交表示（UDR）将图$G$的每个顶点表示为平面中的单位圆盘，使得两个圆盘相交当且仅当它们的顶点在$G$中相邻。具有相邻不相交圆盘的UDR称为单位圆盘接触表示（UDC）。我们证明了判定外平面图或嵌入树是否存在UDR是NP-困难的。我们进一步提供了一个线性时间可判定的卡特彼勒图，承认一个UDR的表征。最后，我们表明，它可以决定在线性时间，如果龙虾图承认一个弱的UDC，它允许非相邻顶点的磁盘之间的交叉。
英文摘要：
A unit disk intersection representation (UDR) of a graph $G$ represents each vertex of $G$ as a unit disk in the plane, such that two disks intersect if and only if their vertices are adjacent in $G$. A UDR with interior-disjoint disks is called a unit disk contact representation (UDC). We prove that it is NP-hard to decide if an outerplanar graph or an embedded tree admits a UDR. We further provide a linear-time decidable characterization of caterpillar graphs that admit a UDR. Finally we show that it can be decided in linear time if a lobster graph admits a weak UDC, which permits intersections between disks of non-adjacent vertices.
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