加权圆盘接触图的识别
Recognizing Weighted Disk Contact Graphs
摘要：
圆盘接触表示通过将顶点双射映射到平面中的相邻不相交圆盘来实现图，使得当且仅当对应的顶点在图中相邻时，两个圆盘彼此接触。判定一个点权平面图是否能实现使得圆盘的半径与点权重合是NP-困难的。在这项工作中，我们通过分析特殊的图形类的问题，减少硬度和易处理性之间的差距。我们发现，它仍然是NP-难的外平面图与单位权重和任意权重的明星，加强以前的硬度结果。在积极的一面，我们提出了建设性的线性时间识别算法的毛毛虫单位重量和嵌入式恒星任意重量。
英文摘要：
Disk contact representations realize graphs by mapping vertices bijectively to interior-disjoint disks in the plane such that two disks touch each other if and only if the corresponding vertices are adjacent in the graph. Deciding whether a vertex-weighted planar graph can be realized such that the disks' radii coincide with the vertex weights is known to be NP-hard. In this work, we reduce the gap between hardness and tractability by analyzing the problem for special graph classes. We show that it remains NP-hard for outerplanar graphs with unit weights and for stars with arbitrary weights, strengthening the previous hardness results. On the positive side, we present constructive linear-time recognition algorithms for caterpillars with unit weights and for embedded stars with arbitrary weights.
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