Determine the immediate successors of three known equivalence classes

Determine the elements immediately after the equivalence classes $\{C_3\}$, $\{P_5\}$, and $\{K_{1,4},K^+_{1,4}\}$ in the partial order induced by the rainbow-forbidden-subgraph preorder.

Background

The paper proves that the equivalence classes {K1,3,K1,3+}\{K_{1,3},K^+_{1,3}\} form the minimum element of the induced partial order and determines the elements immediately after this minimum: {P5}\{P_5\}, {C3}\{C_3\}, and {K1,4,K1,4+}\{K_{1,4},K^+_{1,4}\}.

The authors note that these results provide only limited information about the global structure of the partial order. They therefore ask for the elements that cover, or immediately follow, each of the three specified classes.

References

Determine the elements immediately after ${C_3}$, ${P_5}$ and ${K_{1,4}, K+_{1,4}}$.

— Preorder induced by rainbow forbidden subgraphs  (2502.00667 - Maezawa et al., 2 Feb 2025) in Section 6, “Concluding Remarks,” paragraph immediately before the second displayed Problem