Minimum length word-representants of graph products

Abstract: 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.