  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.

Discrete Mathematics