# Computing the hull number in toll convexity

## TL;DR

This paper introduces a polynomial-time algorithm to compute the toll hull number in general graphs, addressing a specific convexity measure based on tolled walks.

## Contribution

It presents the first polynomial-time algorithm for determining the toll hull number in any graph, advancing understanding of toll convexity.

## Key findings

- Algorithm efficiently computes toll hull number
- Polynomial-time complexity achieved for general graphs
- Enhances analysis of toll convexity in graph theory

## Abstract

A walk $W$ between vertices $u$ and $v$ of a graph $G$ is called a {\em tolled walk between $u$ and $v$} if $u$, as well as $v$, has exactly one neighbour in $W$. A set $S \subseteq V(G)$ is {\em toll convex} if the vertices contained in any tolled walk between two vertices of $S$ are contained in $S$. The {\em toll convex hull of $S$} is the minimum toll convex set containing~$S$. The {\em toll hull number of $G$} is the minimum cardinality of a set $S$ such that the toll convex hull of $S$ is $V(G)$. The main contribution of this work is an algorithm for computing the toll hull number of a general graph in polynomial time.