We introduce the inverse Voronoi diagram problem in graphs: given a graph G with positive edge-lengths and a collection $${\mathbb {U}}$$ U of subsets of vertices of V ( G ), decide whether $${\mathbb {U}}$$ U is a Voronoi diagram in G with respect to the shortest-path metric. We show that the problem is NP-hard, even for planar graphs where all the edges have unit length. We also study the parameterized complexity of the problem and show that the problem is W[1]-hard when parameterized by the number of Voronoi cells or by the pathwidth of the graph.

中文翻译：

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图中的逆 Voronoi 问题 I：硬度

我们在图中引入了逆 Voronoi 图问题：给定一个具有正边长的图 G 和 V ( G ) 顶点子集的集合 $${\mathbb {U}}$$ U，决定 $${\ mathbb {U}}$$ U 是 G 中关于最短路径度量的 Voronoi 图。我们表明该问题是 NP-hard 问题，即使对于所有边都具有单位长度的平面图也是如此。我们还研究了问题的参数化复杂性，并表明当通过 Voronoi 单元的数量或图的路径宽度进行参数化时，该问题是 W[1]-hard。