Computing the hull number in toll convexity

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.