图的同构、色精化和紧性
Graph Isomorphism, Color Refinement, and Compactness
摘要：
颜色细化是一种经典的技术，用于证明两个给定的图G和H是不同构的;它是非常有效的，尽管它并不是在所有图上都成功。我们称一个图G服从色加细，如果它成功地区分G与任何非同构图H。Tinhofer（1991）探索了一种线性规划方法来解决图同构，并定义了紧图：如果一个图的分数自同构多面体是整数，则该图是紧的。丁霍夫指出，紧凑图的同构测试可以通过线性规划非常有效地完成。然而，在多项式时间内刻画和识别紧图的问题仍然是一个悬而未决的问题。我们的结果总结如下：-我们表明，顺从的图是可识别的时间O（（n + m）logn），其中n和m表示的顶点数和边数的输入图。- 我们证明了所有顺从图都是紧的。- 我们研究了Tinhofer和Godsil介绍的相关组合和代数图性质。相应的类的图形形成一个层次结构，我们证明，认识到这些图类中的每一个是P-困难的。特别地，这给出了识别紧致图的第一复杂性下界。
英文摘要：
Color refinement is a classical technique used to show that two given graphs G and H are non-isomorphic; it is very efficient, although it does not succeed on all graphs. We call a graph G amenable to color refinement if it succeeds in distinguishing G from any non-isomorphic graph H. Tinhofer (1991) explored a linear programming approach to Graph Isomorphism and defined compact graphs: A graph is compact if its fractional automorphisms polytope is integral. Tinhofer noted that isomorphism testing for compact graphs can be done quite efficiently by linear programming. However, the problem of characterizing and recognizing compact graphs in polynomial time remains an open question. Our results are summarized below: - We show that amenable graphs are recognizable in time O((n + m)logn), where n and m denote the number of vertices and the number of edges in the input graph. - We show that all amenable graphs are compact. - We study related combinatorial and algebraic graph properties introduced by Tinhofer and Godsil. The corresponding classes of graphs form a hierarchy and we prove that recognizing each of these graph classes is P-hard. In particular, this gives a first complexity lower bound for recognizing compact graphs.
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