In this article we study the right-angled Artin subgroups of a given right-angled Artin group. Starting with a graph Γ, we produce a new graph through a purely combinatorial procedure, and call it the extension graph Γ~e of Γ. We produce a second graph Γ_k~e, the clique graph of Γ~e, by adding an extra vertex for each complete subgraph of Γ~e. We prove that each finite induced subgraph A of Γ~e gives rise to an inclusion A(Λ) → A(Γ). Conversely, we show that if there is an inclusion A(Λ) → A(Γ) then A is an induced subgraph of Γ_k~e. These results have a number of corollaries. Let P_4 denote the path on four vertices and let C_n denote the cycle of length n. We prove that A(P_4) embeds in A(Γ) if and only if P_4 is an induced subgraph of Γ. We prove that if F is any finite forest then A(F) embeds in A(P_4). We recover the first author's result on co-contraction of graphs, and prove that if Γ has no triangles and A(Γ) contains a copy of A(C_n) for some n ≥ 5, then Γ contains a copy of C_m for some 5 ≤ m ≤ n. We also recover Kambites' Theorem, which asserts that if A(C_4) embeds in A(Γ) then Γ contains an induced square. We show that whenever Γ is triangle-free and A(Λ) < A(Γ) then there is an undistorted copy of A(Λ) in A(Γ). Finally, we determine precisely when there is an inclusion A(C_m) → A(C_n) and show that there is no "universal" two-dimensional right-angled Artin group.
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