正则图覆盖的数学性质
Algorithmic Aspects of Regular Graph Covers
摘要：
图$G$覆盖图$H$，如果存在从$G$到$H$的局部双射同态。我们处理正则覆盖，其中该同态由$\textrm{Aut}(G)$的半正则子群的作用规定。我们研究计算方面的定期覆盖，以前没有得到解决。判定问题RegularCover要求给定图$G$和$H$是否$G$规则地覆盖$H$。当$|H|=1$，这个问题成为凯莱图识别的复杂性仍然没有得到解决。当$|G| = |H|$成为图同构问题时，另一种特殊情况出现。我们的主要结果是一个涉及FPT算法求解正则覆盖平面输入$G$在时间$O^*(2^{e(H)/2})$其中$e(H)$表示的边缘数$H$。该算法是基于动态规划，并采用相关的结构文件中证明的理论结果。此外，当$G$是3-连通的，$H$是2-连通的或比率$|G|/|H|$是奇数时，我们可以在多项式时间内解决问题RegularCover。相比之下，Bílka等人（2011）证明了当$H$是一个小的固定图，如$K_4$或$K_5$时，测试平面输入$G$的一般图覆盖是NP完全的。
英文摘要：
A graph $G$ covers a graph $H$ if there exists a locally bijective homomorphism from $G$ to $H$. We deal with regular covers where this homomorphism is prescribed by the action of a semiregular subgroup of $\textrm{Aut}(G)$. We study computational aspects of regular covers that have not been addressed before. The decision problem RegularCover asks for given graphs $G$ and $H$ whether $G$ regularly covers $H$. When $|H|=1$, this problem becomes Cayley graph recognition for which the complexity is still unresolved. Another special case arises for $|G| = |H|$ when it becomes the graph isomorphism problem. Our main result is an involved FPT algorithm solving RegularCover for planar inputs $G$ in time $O^*(2^{e(H)/2})$ where $e(H)$ denotes the number of edges of $H$. The algorithm is based on dynamic programming and employs theoretical results proved in a related structural paper. Further, when $G$ is 3-connected, $H$ is 2-connected or the ratio $|G|/|H|$ is an odd integer, we can solve the problem RegularCover in polynomial time. In comparison, Bílka et al. (2011) proved that testing general graph covers is NP-complete for planar inputs $G$ when $H$ is a small fixed graph such as $K_4$ or $K_5$.
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