Let $G$ be an unlabeled planar and simple $n$-vertex graph. Unlabeled graphs are graphs where the label-information is either not given or lost during the construction of data-structures. We present a succinct encoding of $G$ that provides induced-minor operations, i.e., edge contractions and vertex deletions. Any sequence of such operations is processed in $O(n)$ time in the word-RAM model. At all times the encoding provides constant time (per element output) neighborhood access and degree queries. Optional hash tables extend the encoding with constant expected time adjacency queries and edge-deletion (thus, all minor operations are supported) such that any number of edge deletions are computed in $O(n)$ expected time. Constructing the encoding requires $O(n)$ bits and $O(n)$ time. The encoding requires $\mathcal{H}(n) + o(n)$ bits of space with $\mathcal{H}(n)$ being the entropy of encoding a planar graph with $n$ vertices. Our data structure is based on the recent result of Holm et al. [ESA 2017] who presented a linear time contraction data structure that allows to maintain parallel edges and works for labeled graphs, but uses $\Theta(n \log n)$ bits of space. We combine the techniques used by Holm et al. with novel ideas and the succinct encoding of Blelloch and Farzan [CPM 2010] for arbitrary separable graphs. Our result partially answers the question raised by Blelloch and Farzan whether their encoding can be modified to allow modifications of the graph. As a simple application of our encoding, we present a linear time outerplanarity testing algorithm that uses $O(n)$ bits of space.

翻译：设 $G$ 为无标号简单平面 $n$ 顶点图。无标号图指数据构造过程中标签信息缺失或未被赋予的图。本文提出 $G$ 的便捷编码，支持诱导子式运算（即边收缩与顶点删除）。在字RAM模型下，任意此类运算序列均可在 $O(n)$ 时间内完成。编码全程支持常数时间（每元素输出）邻域访问与度数查询。通过可选哈希表，编码可额外支持常数期望时间邻接查询与边删除（从而支持所有子式运算），且任意数量的边删除可在 $O(n)$ 期望时间内完成。编码构造需 $O(n)$ 比特空间与 $O(n)$ 时间。编码空间复杂度为 $\mathcal{H}(n) + o(n)$ 比特，其中 $\mathcal{H}(n)$ 为编码 $n$ 顶点平面图的熵值。本数据结构基于 Holm 等人 [ESA 2017] 的最新成果——该工作提出线性时间收缩数据结构，可维护平行边并支持标号图，但需 $\Theta(n \log n)$ 比特空间。我们将 Holm 等人的技术与新思路及 Blelloch 与 Farzan [CPM 2010] 的任意可分平面图便捷编码相结合。本结果部分回答了 Blelloch 与 Farzan 提出的问题：其编码能否经修改以支持图操作。作为编码的简单应用，本文提出空间复杂度为 $O(n)$ 比特的线性时间外平面性测试算法。