# A general framework for path convexities

## TL;DR

This paper introduces a comprehensive framework for path convexities in graphs, unifying various existing types and enabling systematic study and new convexity definitions.

## Contribution

It proposes a general framework for path convexities, allowing for systematic analysis and the creation of new convexities beyond existing ones.

## Key findings

- Most existing path convexities are special cases of the framework
- The framework facilitates systematic algorithmic study
- It enables the definition of new, unexplored convexities

## Abstract

In this work we deal with the so-called path convexities, defined over special collections of paths. For example, the collection of the shortest paths in a graph is associated with the well-known geodesic convexity, while the collection of the induced paths is associated with the monophonic convexity; and there are many other examples. Besides reviewing the path convexities in the literature, we propose a general path convexity framework, of which most existing path convexities can be viewed as particular cases. Some benefits of the proposed framework are the systematization of the algorithmic study of related problems and the possibility of defining new convexities not yet investigated.