Finite subgraphs of an extension graph

Let $\Gamma$ be a finite graph and let $\Gamma^{\mathrm{e}}$ be its extension graph. We inductively define a sequence $\{\Gamma_i\}$ of finite induced subgraphs of $\Gamma^{\mathrm{e}}$ through successive applications of an operation called "doubling along a star". Then we show that every finite induced subgraph of $\Gamma^{\mathrm{e}}$ is isomorphic to an induced subgraph of some $\Gamma_i$.
Comment: 8 pages. This note serves as an appendix to the first two authors' paper "Embeddability between right-angled Artin groups" (Geom. Topol. 17 (2013), 493-530) which strengthens and fills in the omitted details of Lemma 3.1 of that paper