The \( P_3 \) intersection graph \( P_3(G) \) of a graph \( G \) is the intersection graph of all induced \( 3 \)-paths in \( G \). In this paper, we prove that any \( P_3 \)-convergent graph is \( P_3^n(G) \)-complete for some \( n \geq 1 \). Additionally, we prove that there are no \( P_3 \)-fixed graphs. The touching number, periodicity, and connectivity of \( P_3(G) \) are also studied.