An adjacency labeling scheme based on a tree-decomposition

TL;DR

This paper introduces a new adjacency labeling scheme for graphs using tree-decomposition, achieving efficient encoding with labels of size proportional to clique-width and logarithmic in the number of vertices.

Contribution

It presents a novel tree-decomposition based encoding scheme for adjacency labeling, improving label size bounds for graphs with bounded clique-width.

Findings

Label size of $O(k \, log k \, log n)$ bits for graphs with clique-width k

Extension of the scheme to certain $k$-probe graph families

Efficient encoding-decoding scheme based on tree-decomposition

Abstract

In this paper we look at the problem of adjacency labeling of graphs. Given a family of undirected graphs the problem is to determine an encoding-decoding scheme for each member of the family such that we can decode the adjacency information of any pair of vertices only from their encoded labels. Further, we want the length of each label to be short (logarithmic in $n$, the number of vertices) and the encoding-decoding scheme to be computationally efficient. We proposed a simple tree-decomposition based encoding scheme and used it give an adjacency labeling of size $O(k \log k \log n)$-bits. Here $k$ is the clique-width of the graph family. We also extend the result to a certain family of $k$-probe graphs.