Improved Version of Short note: Exploring Ideals in Graph Theory

This paper reconsiders the findings of [21] in light of the results presented in [1]. A graph parameter is a numerical concept that characterizes structural aspects of a graph, and numerous such parameters have been studied. Tree-width is one of the most well-known graph width parameters. In this study, we re-evaluate the concept of an ideal within the framework of graph theory. The notion of an ideal is well-established in topology and algebra, where it has been extensively investigated by many researchers due to its deep mathematical importance (see references [2–6]). Here, we explore the relationship between tree-width and ideals on graphs. This investigation contributes to a more profound understanding of these concepts and their roles across various mathematical and logical domains.