Eulerian-spanning set and coboundary operator: An investigation of maxcut beyond planar graphs

2026-05-30

AI Summary

This work extends the MaxCut problem from planar graphs to a broader class of graphs with non-negative edge weights. By introducing a novel integration of Eulerian-spanning sets and the coboundary operator, it overcomes the limitation of Hadlock’s classical approach, which applies only to planar graphs. Building on this theoretical framework, the authors devise a fixed-parameter tractable (FPT) algorithm for graphs with crossing number $k$, achieving a time complexity of $O(2^k (n + k)^{3/2} \log(n + k))$. This algorithm attains state-of-the-art performance for non-negatively weighted instances and offers new structural insights into graph classes such as $k$-contraction apex graphs.

Abstract

Using the concepts of Eulerian-spanning set and coboundary operator, we generalize Hadlock's conversion of the maxcut problem on planar graphs to one on general graphs with non-negative weights. Using our conversion, we can explore algorithms for maxcut beyond the class of planar graphs. We obtain a Fixed-Parameter Tractable algorithm for $k$-contraction apex graphs. Specifically, our algorithm can be applied to graphs with crossing number $k$, giving an $O(2^k(n+k)^{3/2}\log (n+k))$-time algorithm that matches the best known results when restricted to non-negative weights.

Problem

• maxcut
• planar graphs
• non-planar graphs
• crossing number
• k-contraction apex graphs

Innovation

• Eulerian-spanning set
• coboundary operator
• maxcut
• fixed-parameter tractable
• crossing number