Counting Independent Sets in Graphene-like Graphs with Asymmetries Through Hamiltonian Traversals and Minimal Induced Pathwidth

Symmetry plays a fundamental role in the structural analysis of lattice-based systems, particularly in graphene-like molecular structures. In chemical graph theory, counting independent sets is equivalent to computing the Merrifield–Simmons (M–S) index, a key descriptor of molecular stability in conjugated systems. Most existing exact counting methods rely on regular lattice symmetry, where structural uniformity simplifies computation; however, these approaches are difficult to extend to irregular graphs, where symmetry breaking introduces non-local dependencies and increases computational complexity. This paper proposes an asymmetry-aware algorithmic framework based on Hamiltonian traversals and a traversal-induced pathwidth parameter $w\left(G\right)$, defined through backward dependencies. Our method organizes non-local adjacencies into a bounded set of structured constraints, enabling a dynamic programming scheme over a reduced state space. The resulting algorithm runs in time $\mathcal{O}\left(2^{w\left(G\right)}·\mathrm{poly}\left(n\right)\right)$ and is fixed-parameter tractable with respect to $w\left(G\right)$. The results demonstrate that asymmetry-aware traversal strategies enable efficient exact enumeration in irregular mesh graph families, providing a robust computational framework for analyzing molecular descriptors in graphene-based structures with topological defects such as Stone–Wales transformations.