Succinct Planar Encoding with Minor Operations

Let $G$ be an unlabeled planar and simple $n$-vertex graph. {We present a succinct encoding of $G$ that provides induced-minor operations, i.e., edge contraction and vertex deletions. Any sequence of such operations is processed in $O(n)$ time.} In addition, the encoding provides constant time per element neighborhood access and degree queries. Using optional hash tables the encoding additionally provides constant {expected} time adjacency queries as well as an edge-deletion operation {(thus, all minor operations are supported)} such that any number of such 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 $O(n \log n)$ 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 if their encoding can be modified to allow modifications of the graph.