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.

翻译：一个图$G = (V, E)$被称为单词可表示的，如果存在一个由字母表$V$中的字母构成的单词$w$，使得对于任意一对顶点$x$和$y$，$xy \in E$当且仅当$x$和$y$在$w$中交替出现。Gaetz和Ji最近引入了单词可表示图的最小长度单词表示概念。他们确定了某些图类（如树和圈）的单词表示可能的最小长度。已知笛卡尔积和根积保持单词可表示性。此外，Broere利用基于出现的函数构建了一个统一单词来表示$G$与$K_n$的笛卡尔积。在本文中，我们分别利用同态和基于出现的函数研究笛卡尔积和根积的最小长度单词表示。同时，我们解决了Broere在其硕士论文中提出的一个开放问题，该问题要求为两个任意单词可表示图的笛卡尔积构建一个单词。