For a finite graph Γ, let G(Γ) be the right-angled Artin group defined by the complement graph of Γ. We show that, for any linear forest Λ and any finite graph Γ, G(Λ) can be embedded into G(Γ) if and only if Λ can be realised as a full subgraph of Γ. We also prove that if we drop the assumption that Λ is a linear forest, then the above assertion does not hold, namely, for any finite graph Λ, which is not a linear forest, there exists a finite graph Γ such that G(Λ) can be embedded into G(Γ), though Λ cannot be embedded into Γ as a full subgraph.2010 Mathematics Subject Classification. 20F36 (primary).

In this article, we determine the function (Sg,p) such that the right-angled Artin group G(Pm) is embedded in the mapping class group Mod(Sg,p) if and only if m is not more than (Sg,p). Using this function and Birman-Hilden theory, we prove that Mod(S 0,p ) is virtually embedded in Mod(S g,0 ) if and only if p ≤ 2g + 2.

Abstract. In this article, we prove that embeddings of right-angled Artin group A 1 on the complement of a linear forest into another right-angled Artin group A 2 can be reduced to full embeddings of the defining graph of A 1 into the extension graph of the defining graph of A 2 .

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