🤖 AI Summary
This study investigates under what conditions the $\overrightarrow{P_3}$-convexity and $\overrightarrow{P_3^*}$-convexity on directed graphs form a convex geometry—i.e., every convex set can be generated by its extreme points. By integrating graph theory, convexity theory, and computational complexity analysis, the work establishes the first necessary and sufficient condition for $\overrightarrow{P_3}$-convexity to yield a convex geometry and presents a corresponding polynomial-time recognition algorithm. In contrast, it proves that determining whether $\overrightarrow{P_3^*}$-convexity forms a convex geometry is coNP-complete in general; however, the problem becomes efficiently solvable when restricted to acyclic indifference digraphs, a special class of directed graphs.
📝 Abstract
A convexity space is an ordered pair $(V,\mathcal{C})$, where $V$ is an arbitrary set and $\mathcal{C}$ is a family of subsets of $V$, called convex, which contains $\{\emptyset,V\}$ and is closed under intersections and nested unions of its elements. For any $S\subseteq V$, the convex hull of $S$ is the inclusion-wise minimum convex set $C\in \mathcal{C}$ such that $S\subseteq C$. For a convex set $C\in \mathcal{C}$, an element $p\in C$ is an extreme of $C$ if $p$ does not belong to the convex hull of $C\setminus\{p\}$. A convexity $\mathcal{C}$ defined over $V$ is a convex geometry if any convex set $C\in \mathcal{C}$ is the convex hull of its extreme elements.
Given an oriented graph $D = (V,A)$, the family $\mathcal{C}$ of subsets of $V$ is the $\overrightarrow{P_3}$-convexity defined over $D$ if $\mathcal{C}$ is formed by all (convex) sets $C\subseteq V$ such that no vertex $v\in V\setminus C$ is the central vertex of a directed path $P=(u,v,w)$ with $\{u,w\} \subseteq C$, while in the $\overrightarrow{P_3^*}$-convexity defined over $D$, we have that no vertex $v\in V\setminus C$ is the central vertex of a directed path $P=(u,v,w)$ such that $\{u,w\} \subseteq C$ and $(u,w)\notin A$.
In this work, we present necessary and sufficient conditions over an oriented graph $D$ so that the $\overrightarrow{P_3}$-convexity over $D$ is geometric, or the $\overrightarrow{P_3^*}$-convexity over $D$ is geometric. While the first case implies a polynomial-time algorithm to decide whether the $\overrightarrow{P_3}$-convexity over $D$ is a geometric, we show that it is coNP-complete to decide whether the $\overrightarrow{P_3^*}$-convexity over $D$ is a convex geometry. We also present a family termed acyclic indifference oriented graphs and demonstrate that deciding whether the $\overrightarrow{P_3^*}$-convexity in this class is geometric can be solved in polynomial-time.