A Study Into the Relationship Between Treewidth and Pathwidth

We explain the tree-and-pathwidth of a simple, connected, undirectedgraph. We then exemplify their different use cases by showing how theycan be applied to solve different famously hard graph theoretic problems,and comparisons are made to evaluate these parameters. In the end wefind that both parameters have their uses, and that interestingly they bothcan be alternatively characterised, beyond their respective decompositions.