NP-completeness of (r+1)-Role Assignment on complementary prisms for r ≥ 3

Prove that for every integer r ≥ 3, the decision problem of whether the complementary prism G\overline{G} of an input graph G admits an (r+1)-role assignment is NP-complete.

Background

Role assignment asks whether a graph admits a vertex mapping to r roles such that vertices with the same role have identical multisets of neighboring roles; the induced role graph connects roles that co-occur on adjacent vertices. The general r-Role Assignment problem is NP-complete for any fixed r ≥ 3 on arbitrary graphs.

This paper proves that deciding 3-role assignments on complementary prisms G\overline{G} is polynomial-time solvable. Prior work showed NP-completeness for complementary prisms when the role graph is specifically K'_{1,r} and, motivated by this, proposed a broader conjecture asserting NP-completeness for (r+1)-Role Assignment on complementary prisms for all r ≥ 3. Establishing this conjecture would extend the present paper’s tractable boundary at r=3 to a hardness threshold for higher numbers of roles.

References

In this sense,, considered the role graph $K'{1,r}$ which is the bipartite graph $K{1,r}$ with a loop at the vertex of degree $r$ and showed that the problem of deciding whether a prism complement has a $(r+1)$-role assignment, when the role graph is $K'_{1,r}$, is $\NP$-complete and set the conjecture that, for $r\geq 3$, $(r+1)$-Role Assignment for complementary prisms is $\NP$-complete.

— Computing a 3-role assignment is polynomial-time solvable on complementary prisms  (2402.06068 - Castonguay et al., 2024) in Section 1 (Introduction)