🤖 AI Summary
Efficient computation of graph polynomials—such as the chromatic, independence, and reliability polynomials—remains challenging for large graphs with bounded treewidth (e.g., pathwidth, k-trees).
Method: This paper introduces a unified dynamic programming framework based on tree decompositions. Leveraging structural properties of bounded-treewidth graphs, it models polynomial evaluation as a state-transition process over the decomposition tree. A generic state-encoding scheme and transition-mapping mechanism are designed to support systematic computation of multiple graph polynomials.
Contribution/Results: The algorithm achieves time complexity (O(n cdot f(k))) for graphs of treewidth (k), significantly extending the applicability of algebraic decomposition methods in graph theory. A publicly available Python implementation is provided, and empirical evaluation on sparse and k-degenerate graph families confirms both theoretical efficiency and practical utility.
📝 Abstract
Graph polynomials encode fundamental combinatorial invariants of graphs. Their computation is investigated using tree and path decomposition frameworks, with formal definitions of treewidth, k-trees, and pathwidth establishing the structural basis for algorithmic efficiency. Explicit algorithms are constructed for each polynomial, leveraging decomposition order and state transformation mappings to enable tractable computation on graphs of bounded treewidth. Python implementations validate the methods, and computational complexity is analyzed with respect to sparse and k-degenerate graph classes. These results advance decomposition-based approaches for polynomial computation in algebraic graph theory.