Determine optimal tile and bond-edge counts for arbitrary graphs

Determine the values of the minimum bond-edge-type count $B_3(G)$ and minimum tile-type count $T_3(G)$ for arbitrary target graphs under the Scenario 3 constraint, and construct valid optimal pots for such graphs.

Background

For a target graph G, the quantities B_3(G) and T_3(G) measure the minimum numbers of bond-edge types and tile types, respectively, needed for a pot that realizes G without realizing smaller graphs or nonisomorphic graphs of the same order. The paper develops lower bounds for unswappable graphs and upper-bound constructions for certain regular graphs, but it explicitly states that the general values and corresponding constructions remain unknown.

References

In particular, the values of $B_3(G)$ and $T_3(G)$ are not known for arbitrary graphs (not to mention how to construct a valid pot for a given graph).

— Optimal Constructions for DNA Self-Assembly of $k$-Regular Graphs  (2502.03716 - Baek et al., 6 Feb 2025) in Section 1, subsection “Research Directions”