All-path convexity: Combinatorial and complexity aspects

Let $\mathrm{\P }$ be any collection of paths of a graph $G=(V,E)$. For $S\subseteq V$, define $I(S)=S\cup \{v\mid vliesinapathof\mathrm{\P }withendpointsinS\}$. Let $\text{C}$ be the collection of fixed points of the function $I$, that is, $\text{C}=\{S\subseteq V\mid I(S)=S\}$. It is well known that $(V,\text{C})$ is a finite convexity space, where the members of $\text{C}$ are precisely the convex sets. If $\mathrm{\P }$ is taken as the collection of all the paths of $G$, then $(V,\text{C})$ is the {\em all-path convexity} with respect to graph $G$. In this work we study how important parameters and problems in graph convexity are solved for the all-path convexity.