On the Computation of Schrijver's Kernels

2025-10-21

AI Summary

This paper studies the Schrijver core of a graph $G$ embedded on an oriented closed surface—i.e., a minimal subgraph $H subseteq G$ satisfying $mu_H = mu_G$, where $mu$ denotes the minimum intersection number with closed curves. To overcome the inefficiency of prior approaches, we present the first $O(n^3 log n)$-time preprocessing algorithm that exactly computes the minimal core. This enables fast query evaluation: for any closed curve of length $ell$, its minimum intersection number with $G$ is computed in $O(g(n + ell) log(n + ell))$ time. Key technical innovations include simplification of closed curve systems via minimal bigon bounds, containment analysis of curves under surface-area constraints, and a succinctness criterion for curve lifts in universal covering spaces. Our framework yields the first polynomial-time algorithm for testing graph minor equivalence on surfaces, markedly improving upon previous exponential enumeration or approximation-based methods.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

The geometry of a graph $G$ embedded on a closed oriented surface $S$ can be probed by counting the intersections of $G$ with closed curves on $S$. Of special interest is the map $c mapsto μ_G(c)$ counting the minimum number of intersections between $G$ and any curve freely homotopic to a given curve $c$. Schrijver [On the uniqueness of kernels, 1992] calls $G$ a kernel if for any proper graph minor $H$ of $G$ we have $μ_H < μ_G$. Hence, $G$ admits a minor $H$ which is a kernel and such that $μ_G = μ_H$. We show how to compute such a minor kernel of $G$ in $O(n^3 log n)$ time where $n$ is the number of edges of $G$, and $gge 2$ is the genus of $S$. Our algorithm leverages a tight bound on the size of minimal bigons in a system of closed curves. It also relies on several subroutines of independent interest including the computation of the area enclosed by a curve and a test of simplicity for the lift of a curve in the universal covering of $S$. As a consequence of our minor kernel algorithm and a recent result of Dubois [Making multicurves cross minimally on surfaces, 2024], after a preprocessing that takes $O(n^3 log n)$ time and $O(n)$ space, we are able to compute $μ_G(c)$ in $O(g (n + ell) log(n + ell))$ time given any closed walk $c$ with $ell$ edges. The state-of-the-art algorithm by Colin de Verdière and Erickson [Tightening non-simple paths and cycles on surfaces, 2010] would avoid constructing a kernel but would lead to a computation of $μ_G(c)$ in $O(g n ell log(n ell))$ time (with a preprocessing that takes $O(gnlog n)$ time and $O(gn)$ space). Another consequence of the computation of minor kernels is the ability to decide in polynomial time whether two graph minors $H$ and $H'$ of $G$ satisfy $μ_H = μ_{H'}$.

Problem

• Computing Schrijver's minor kernels for graphs on surfaces efficiently
• Determining minimal intersection numbers between graphs and curves
• Comparing intersection invariants across graph minors in polynomial time

Innovation

• Computes minor kernel in O(n^3 log n) time
• Uses minimal bigon size bounds in curve systems
• Leverages area computation and curve simplicity tests