arXiv 1503.065562015-03math.CO

3-正则图覆盖的连通约简

3-connected Reduction for Regular Graph Covers

Jiří Fiala, Pavel Klavík, Jan Kratochvíl, Roman Nedela

备注：: The journal version of the first part of arXiv:1402.3774

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，其中这个同态由$\textrm{Aut}(G)$的半正则子群$Γ$的作用规定;因此$H \cong G / Γ$。本文研究了$G$中正则图关于1-割和2-割的覆盖行为。我们描述的减少，产生一系列的图形$G = G_0,\dots,G_r$，使$G_{i+1}$创建从$G_i$通过替换某些包含最小的子图与着色的边缘。该过程以3-连通或循环或$K_2$的本原图$G_r$结束。这种归约可以被看作是对Mac Lane（1937）、Trachtenbrot（1958）、Tutte（1966）、Hopcroft和Tarjan（1973）、Cuningham和Edmonds（1980）、Walsh（1982）和其他人的归约的非平凡修改。我们的方法的一个新的特点是，在每一步中，所有必要的信息对称的$G$被保存。正则覆盖投影$G_0\to H_0$诱导正则覆盖投影$G_i \to H_i$，其中$H_i$是$H_0$的第$i$个商约简。该属性允许从$G_r$的可能的子项$H_r$构造$G_0$的所有可能的子项$H_0$。将此方法应用于平面图，给出了Negami定理（1988）的一个证明.我们的结构性结果也被用在随后的论文中，当$G$是平面图时，用于正则覆盖测试，以及平面图的自同构群的归纳特征（参见Babai（1973））。

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 coverings in which this homomorphism is prescribed by an action of a semiregular subgroup $Γ$ of $\textrm{Aut}(G)$; so $H \cong G / Γ$. In this paper, we study the behaviour of regular graph covering with respect to 1-cuts and 2-cuts in $G$. We describe reductions which produce a series of graphs $G = G_0,\dots,G_r$ such that $G_{i+1}$ is created from $G_i$ by replacing certain inclusion minimal subgraphs with colored edges. The process ends with a primitive graph $G_r$ which is either 3-connected, or a cycle, or $K_2$. This reduction can be viewed as a non-trivial modification of reductions of Mac Lane (1937), Trachtenbrot (1958), Tutte (1966), Hopcroft and Tarjan (1973), Cuningham and Edmonds (1980), Walsh (1982), and others. A novel feature of our approach is that in each step all essential information about symmetries of $G$ are preserved. A regular covering projection $G_0\to H_0$ induces regular covering projections $G_i \to H_i$ where $H_i$ is the $i$-th quotient reduction of $H_0$. This property allows to construct all possible quotients $H_0$ of $G_0$ from the possible quotients $H_r$ of $G_r$. By applying this method to planar graphs, we give a proof of Negami's Theorem (1988). Our structural results are also used in subsequent papers for regular covering testing when $G$ is a planar graph and for an inductive characterization of the automorphism groups of planar graphs (see Babai (1973) as well).

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英文摘要：

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