The injective envelope of simple modules over Leavitt path algebras II: simples arising from exclusive cycles

Let $K$ be any field, $E$ any directed graph, and $L_K(E)$ the associated Leavitt path algebra. As described first by Chen, and subsequently generalized by Ara and Rangaswamy, for each cycle $c$ in $E$ one can build the simple left $L_K(E)$-module $V_c^E$, and then more generally $V_{p(x),c}^E$ (whe