无短圈图的根计算
Computing Graph Roots Without Short Cycles
摘要：
图G是图H的平方，如果两个顶点x，y在G中有一条边当且仅当x，y在H中的距离至多为2。给定H，很容易计算它的平方H2，然而Motwani和Sudan证明了确定给定的图G是否是某个图H（围长为3）的平方是NP-完全的。本文研究了小围长图的平方图的特征和识别问题，即对某个小围长图H，判定G是否= H~2。主要结果如下。- 对于某些围长至少为7的图的平方，有一个图论刻画。一个推论是，如果一个图G有一个周长至少为7的平方根H，那么H在同构下是唯一的。- 有一个多项式时间的算法来识别G是否= H 2的一些图H的围长至少为6。- 对于围长为4的图H，判定G是否= H2是NP-完全的。这些结果几乎提供了一个二分法定理的识别问题的复杂性方面的周长的平方根。算法和图论的结果推广了以前关于树平方根的结果，并提供了计算小围长图平方根的多项式时间算法。还将讨论一些开放的问题和建议。
英文摘要：
Graph G is the square of graph H if two vertices x, y have an edge in G if and only if x, y are of distance at most two in H. Given H it is easy to compute its square H2, however Motwani and Sudan proved that it is NP-complete to determine if a given graph G is the square of some graph H (of girth 3). In this paper we consider the characterization and recognition problems of graphs that are squares of graphs of small girth, i.e. to determine if G = H2 for some graph H of small girth. The main results are the following. - There is a graph theoretical characterization for graphs that are squares of some graph of girth at least 7. A corollary is that if a graph G has a square root H of girth at least 7 then H is unique up to isomorphism. - There is a polynomial time algorithm to recognize if G = H2 for some graph H of girth at least 6. - It is NP-complete to recognize if G = H2 for some graph H of girth 4. These results almost provide a dichotomy theorem for the complexity of the recognition problem in terms of girth of the square roots. The algorithmic and graph theoretical results generalize previous results on tree square roots, and provide polynomial time algorithms to compute a graph square root of small girth if it exists. Some open questions and conjectures will also be discussed.
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