Completing a restricted target with universal clique vertices

The sets $W$ and $U$ are disjoint finite subsets of $V(J)$ with union $V(J)$. Thus $\mathrm{\ell }\le n$ and $n=\mathrm{\ell }+\mathrm{\mid }U\mathrm{\mid }$. By the characterization of $n-\mathrm{\ell }$, $\mathrm{\mid }U\mathrm{\mid }=n-\mathrm{\ell }=\mathrm{\mid }C\mathrm{\mid }$. Choose a bijection $b:U\to C$. This also applies if either set is empty.

Define $f(v)=v$ for $v\in W$ and $f(v)=b(v)$ for $v\in U$. The two domains partition $V(J)$, so each vertex receives exactly one value in $W\cup C=V(G)$; thus this defines a function $f:V(J)\to V(G)$. The two image sets $W$ and $C$ are disjoint. Since the identity on $W$ and $b$ are injective, $f$ is injective; it fixes $W$ and maps $U$ bijectively onto $C$.

Let $\{x,y\}$ be an edge of $J$. If both endpoints lie in $W$, it is an edge of $J[W]$ and hence of $G$. If both lie in $U$, their distinct images lie in $C$ and are adjacent in the complete graph on $C$. If one lies in each part, their images are adjacent by the complete connection. Therefore $f$ preserves every edge of $J$.

The map $f$ is consequently an ordinary embedding, and its image vertices and image edges form an ordinary copy of $J$ in $G$. The construction and checks remain valid when $W$ or $U$ is empty, giving the two stated boundary cases.