A graph G is called a pairwise compatibility graph (PCG, for short) if it admits a tuple $$(T,w, d_{\min },d_{\max })$$ ( T , w , d min , d max ) of a tree T whose leaf set is equal to the vertex set of G , a non-negative edge weight w , and two non-negative reals $$d_{\min }\le d_{\max }$$ d min ≤ d max such that G has an edge between two vertices $$u,v\in V$$ u , v ∈ V if and only if the distance between the two leaves u and v in the weighted tree ( T , w ) is in the interval $$[d_{\min }, d_{\max }]$$ [ d min , d max ] . The tree T is also called a witness tree of the PCG G . How to recognize PCGs is a wide-open problem in the literature. This paper gives a complete characterization for a graph to be a star-PCG (a PCG that admits a star as its witness tree), which provides us the first polynomial-time algorithm for recognizing star-PCGs.

中文翻译：

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表征 Star-PCG

如果图 G 接受元组 $$(T,w, d_{\min },d_{\max })$$ ( T , w , d min , d max ) 的叶子集等于 G 的顶点集、非负边权 w 和两个非负实数 $$d_{\min }\le d_{\max }$$ d min ≤ d max 使得 G 在两个顶点之间有一条边 $$u,v\in V$$ u , v ∈ V 当且仅当加权树 ( T , w ) 中两个叶子 u 和 v 之间的距离为在区间 $$[d_{\min }, d_{\max }]$$ [ d min , d max ] 。树 T 也称为 PCG G 的见证树。如何识别 PCG 是文献中一个广泛存在的问题。这篇论文给出了一个星形PCG（一种承认星形作为其见证树的PCG）的图的完整特征，它为我们提供了第一个用于识别星形PCG的多项式时间算法。