  A graph $G = (V, E)$ is said to be word-representable if a word $w$ can be
formed using the letters of the alphabet $V$ such that for every pair of
vertices $x$ and $y$, $xy \in E$ if and only if $x$ and $y$ alternate in $w$.
Gaetz and Ji have recently introduced the notion of minimum length
word-representants for word-representable graphs. They have also determined the
minimum possible length of the word-representants for certain classes of
graphs, such as trees and cycles. It is know that Cartesian and Rooted products
preserve word-representability. Moreover, Broere constructed a uniform word
representing the Cartesian product of $G$ and $K_n$ using occurrence based
functions.
  In this paper, we study the minimum length of word-representants for
Cartesian and Rooted products using morphism and occurrence based function,
respectively. Also, we solve an open problem posed by Broere in his master
thesis. This problem asks to construct a word for the Cartesian product of two
arbitrary word-representable graphs.
