A generalization of perfect codes in the presence of star multiset transpositions

Let $0<\mathrm{\ell }\in \mathbb{Z}$. The notion of an efficient dominating set or perfect code $S$ of a graph $G$ is generalized to that of an efficient dominating${}^{\mathrm{\ell }}$-set or perfect${}^{\mathrm{\ell }}$code, of the graph $G$, meaning that each vertex $v$ of $V(G)\setminus S$ has exactly $\mathrm{\ell }$ neighbors in $S$, instead of just one neighbor. Such generalization is applied to star $j$-set transposition graphs based on permutations of multisets with each symbol repeated $j$ times, ($j\in \{\mathrm{\ell },\mathrm{\ell }-1\}$). In such vertex-transitive graphs this approach produces total colorings, efficient dominating sets, also called perfect codes, etc.