Counting $P_3$-convex sets in graphs

TL;DR

This paper explores the counting of $P_3$-convex sets in graphs, characterizes extremal graphs, analyzes computational complexity, and develops efficient algorithms for specific graph classes.

Contribution

It characterizes extremal graphs maximizing $P_3$-convex sets, proves counting complexity is $ ext{ extonehalf} ext{ exttrademark}$-complete, and provides linear-time algorithms for trees and threshold graphs.

Findings

Maximizing $P_3$-convex sets in extremal graphs identified.

Counting $P_3$-convex sets is $ ext{ extonehalf} ext{ exttrademark}$-complete on split graphs.

Linear-time algorithms developed for trees and threshold graphs.

Abstract

We study the $P_3$-convexity, the path convexity generated by all three-vertex paths, and focus on the problem of counting the $P_3$-convex vertex sets of a graph $G$, denoted by $\noc(G)$. First, we settle the associated extremal question: we characterize the $n$-vertex graphs maximizing $\noc(G)$ among all graphs and determine the connected extremal graphs. Next, we investigate computational complexity and show that counting $P_3$-convex sets is $\#\mathsf{P}$-complete already on split graphs, even under additional structural restrictions. On the positive side, we identify two tractable subclasses, namely trees and threshold graphs, and obtain linear-time algorithms for both. Finally, we design nontrivial exact exponential-time algorithms for general graphs, combining structural decomposition, propagation rules capturing forced consequences of $P_3$-convexity, and fast counting of independent sets in auxiliary graphs. The resulting strategy becomes particularly effective on graph classes where large independent sets are guaranteed and can be found efficiently.