Counting $P_3$-convex sets in graphs

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.