On the connection between fixed point theorems on metric spaces with graphs and $\mathbb{P}$ sets

The Banach contraction principle is one of the most famous and applied results in recent mathematical history. Due to its utility, plenty of generalizations have been established. One of them considers a contraction principle on metric spaces with graphs, while another confines the contraction principle to pairs of elements inside a $\mathbb{P}$ set, a generalization of partial orders. In this work we examine the similarities of both approaches, establishing the connection between theorems of metric spaces with graphs and metric spaces with $\mathbb{P}$ sets and restating results from one approach to the other and vice versa.