  How to efficiently represent a graph in computer memory is a fundamental data
structuring question. In the present paper, we address this question from a
combinatorial point of view. A representation of an $n$-vertex graph $G$ is
called implicit if it assigns to each vertex of $G$ a binary code of length
$O(\log n)$ so that the adjacency of two vertices is a function of their codes.
A necessary condition for a hereditary class $X$ of graphs to admit an implicit
representation is that $X$ has at most factorial speed of growth. This
condition, however, is not sufficient, as was recently shown in [Hatami &
Hatami, FOCS 2022]. Several sufficient conditions for the existence of implicit
representations deal with boundedness of some parameters, such as degeneracy or
clique-width. In the present paper, we analyze more graph parameters and prove
a number of new results related to implicit representation and factorial
properties.
