Succinct Planar Encoding with Minor Operations

TL;DR

This work achieves a succinct encoding of unlabeled planar graphs that remains dynamic under induced-minor operations (edge contractions and vertex deletions) with $O(n)$ time and $\mathcal{H}(n)+o(n)$ bits, while maintaining constant-time neighborhood and degree queries; optional hashing extends to constant-time adjacency queries and $O(n)$ expected time for edge deletions. The core methodology fuses Holm et al.'s contraction framework for labeled graphs with Blelloch-Farzan's table-based encoding for separable graphs, extended to a dynamic, unlabeled setting via multilevel divisions, dynamic label mappings, and boundary graphs. An explicit dynamic encoding is shown to support arbitrary contractions in $G$ with $O(n)$ total time and space, and is leveraged to obtain a linear-time outerplanarity test in $O(n)$ space. The results advance succinct dynamic graph representations and provide practical, space-efficient algorithms for planarity-related tasks, including outerplanarity testing.

Abstract

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 $Θ(n \log n)$ 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.

Theorems & Definitions (25)

• Theorem 1
• Corollary 2
• Lemma 3
• Lemma 4: holm_et_al:LIPIcs:2017:7875
• Lemma 5
• Lemma 6: 10.1145/1290672.1290680
• Lemma 7
• Corollary 8
• Corollary 9
• Theorem 10