Disconnected Character graphs and odd Dominating sets

Suppose $\mathrm{\Gamma }$ is a finite simple graph. If $D$ is a dominating set of $\mathrm{\Gamma }$ such that each $x\in D$ is contained in the set of vertices of an odd cycle of $\mathrm{\Gamma }$, then we say that $D$ is an odd dominating set for $\mathrm{\Gamma }$. For a finite group $G$, let $\mathrm{\Delta }(G)$ denote the character graph built on the set of degrees of the irreducible complex characters of $G$. In this paper, we show that the complement of $\mathrm{\Delta }(G)$ contains an odd dominating set, if and only if $\mathrm{\Delta }(G)$ is a disconnected graph with non-bipartite complement.