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

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