Shortest Circuits in Homology Classes of Graphs
Ye Luo
TL;DR
The paper addresses the problem of detecting and counting the shortest circuits in prescribed homology classes of graphs, formalized as HSCDP for nonzero $\alpha\in H_1(G,\mathbb{Z})$. It introduces direction-consistent circuits and extends classical Eulerian tools by developing a generalized Hierholzer's algorithm for existence and a generalized BEST theorem for counting, with a weighted Laplacian $L_\alpha(G)$ underpinning the counts. A contraction-based framework solves HSCDP in the connected-case and highlights when the derived circuit length is $\tau_\mu(C)=2\mu(T_\alpha)+\|\alpha\|_\mu$, while noting potential suboptimality in the general, multi-component setting due to Steiner-tree effects. The framework is then applied to the 1-carrier Transportation Routing Problem of type $(1,1)$ via an edge-doubling reduction to a lattice SVP problem on $H_1(\hat{G},\mathbb{Z})$, yielding partial (and in favorable cases optimal) routing solutions, thereby linking graph homology, circuit optimization, and practical routing problems.
Abstract
Recently, the study of circuits and cycles within the homology classes of graphs has attracted considerable research interest. However, the detection and counting of shorter circuits in homology classes, especially the shortest ones, remain underexplored. This paper aims to fill this gap by solving the problem of detecting and counting the shortest cycles in homology classes, leveraging the concept of direction-consistent circuits and extending classical results on Eulerian circuits such as Hierholzer's algorithm and the BEST theorem. As an application, we propose the one-carrier transportation routing problem and relate it to a circuit detection problem in graph homology.