In this paper we define the generalised non-commuting graph Γ(H,K) , where H and K are two subgroups of a non-abelian group G. Take (H ∪ K) \ (CH(K) ∪ CK(H)) as the vertices of the graph and two distinct vertices x and y join, whenever x or y is in H and [x, y] 6= 1. We obtain diameter and girth of this graph. Also, we discuss the dominating set and planarity of Γ(H,K) . Moreover, we try to find a connection between Γ(H,K) and the relative commutativity degree of two subgroups d(H, K). Furthermore, we prove that if Γ(H,G) ∼= Γ(K,G) , then ΓH ∼= ΓK. And finally we introduce a special case when subgroup K is equal to the non-abelian group G.

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