Adjacency Labelling for Planar Graphs (and Beyond) (2003.04280v4)

Abstract: We show that there exists an adjacency labelling scheme for planar graphs where each vertex of an $n$-vertex planar graph $G$ is assigned a $(1+o(1))\log_2 n$-bit label and the labels of two vertices $u$ and $v$ are sufficient to determine if $uv$ is an edge of $G$. This is optimal up to the lower order term and is the first such asymptotically optimal result. An alternative, but equivalent, interpretation of this result is that, for every $n$, there exists a graph $U_n$ with $n^{1+o(1)}$ vertices such that every $n$-vertex planar graph is an induced subgraph of $U_n$. These results generalize to bounded genus graphs, apex-minor-free graphs, bounded-degree graphs from minor closed families, and $k$-planar graphs.

• The paper introduces a near-optimal adjacency labelling scheme for planar graphs, encoding n-vertex graphs in (1+o(1)) log₂ n bits.
• The scheme extends to various graph families, including bounded genus, apex-minor-free, and k-planar graphs, demonstrating broad applicability.
• The methodology leverages graph products and tree-decomposition techniques to enable efficient storage and rapid edge query capabilities.

An Overview of Adjacency Labelling for Planar Graphs (and Beyond)

The paper "Adjacency Labelling for Planar Graphs (and Beyond)" by Vida Dujmović et al. presents a significant advancement in the efficient encoding of structural information about planar graphs and other related families. The central contribution is an adjacency labelling scheme for planar graphs that achieves near-optimal label length while allowing for the adjacency of any two vertices to be determined solely from their associated labels.

Key Contributions

1. Label Length Efficiency: The paper establishes an adjacency labelling scheme for planar graphs where each $n$-vertex graph can be encoded with a $(1+o(1))\log_2 n$-bit label. This label length is optimal up to lower order terms. The scheme extends to several other graph classes, indicating its general applicability.
2. Graph Families: Beyond planar graphs, the scheme applies to other graph families including:
   • Bounded genus graphs
   • Apex-minor-free graphs
   • Minor-closed families with bounded degree
   • $k$-planar graphs

For each of these classes, the scheme provides a similar efficient adjacency labelling.

1. Theoretical Framework: The paper leverages structured insights into the graph product and labelling schemes. Specifically, the work utilizes the strong product of bounded treewidth graphs and paths to structure the approach to labelling.

Technical Approach

The authors build upon graph theoretical concepts of $t$-trees and interval graphs to construct labelling schemes that represent graph adjacency effectively. By representing a planar $n$-vertex graph as a subgraph of a strong product of a treewidth-bounded graph and a path, the authors manage to embed complex graph adjacency relations into concise labels.

Furthermore, the paper introduces operations on binary search trees such as "bulk insertion", "bulk deletion", and "rebalancing," facilitating the progressive construction of a tree structure that allows for efficient encoding and comparison of node positions.

Implications and Future Directions

The implications of achieving an asymptotically optimal adjacency labelling scheme are multifaceted. Practically, such labelling schemes allow for efficient storage and querying of graph edges, beneficial for network design and data structure optimization in constrained environments.

Theoretically, the work hints at potential generalizations to broader families of graphs, specifically those excluding a fixed apex graph as a minor. Additionally, it raises intriguing conjectures, such as generalized labelling schemes for any monotone family of graphs satisfying specific growth conditions in terms of the number of labelled graph instances.

Conclusion

The paper by Dujmović et al. is a significant step in the paper of informative labelling schemes, offering a robust and efficient solution for planar graphs as well as extending beyond to other complex graph families. The methods developed offer promising avenues for further research into graph encoding strategies and their applications in computational and applied graph theory contexts.