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Let $H$ be a subgroup of a finite non-abelian group $G$ and $g \in G$. Let$Z(H, G) = \{x \in H : xy = yx, \forall y \in G\}$. We introduce the graph$\Delta_{H, G}^g$ whose vertex set is $G \setminus Z(H, G)$ and two distinctvertices $x$ and $y$ are adjacent if $x \in H$ or $y \in H$ and $[x,y] \neq g,g^{-1}$, where $[x,y] = x^{-1}y^{-1}xy$. In this paper, we determine whether$\Delta_{H, G}^g$ is a tree among other results. We also discuss about itsdiameter and connectivity with special attention to the dihedral groups.

$g$-noncommuting graph of a finite group relative to its subgroups