A graph G is called to be fully cycle extendable graph [3] if each vertex of G belongs to a triangle and for any cycle C with |V(C)| < |V(G)| there exists a cycle C ′ in G such that V(C) ⊂ V(C ′) and |V(C ′)| = |V(C)|+1. In this paper, we show that every graph G that is triangularly connected, partly claw-free and {K 1,4 , K 4 }-free is fully cycle extendable graph if its claw centers set is P 4-free. This paper generalizes the concept of Hendry fully cycle extendable graph [3] for the largest superclass of partly claw-free graphs defined by Abbas and Benmeziane [1].