Finite subgraphs of an extension graph

Let $\mathrm{\Gamma }$ be a finite graph and let ${\mathrm{\Gamma }}^{\mathrm{e}}$ be its extension graph. We inductively define a sequence $\{{\mathrm{\Gamma }}_i\}$ of finite induced subgraphs of ${\mathrm{\Gamma }}^{\mathrm{e}}$ through successive applications of an operation called “doubling along a star”. Then we show that every finite induced subgraph of ${\mathrm{\Gamma }}^{\mathrm{e}}$ is isomorphic to an induced subgraph of some ${\mathrm{\Gamma }}_i$.