Recognizing hyperelliptic graphs in polynomial time

Jelco M Bodewes,Hans L Bodlaender,Gunther Cornelissen,Marieke Van Der Wegen

doi:10.1016/j.tcs.2020.02.013

Jelco M Bodewes, Hans L Bodlaender + Show 2 more

Open Access

https://doi.org/10.1016/j.tcs.2020.02.013

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Journal: Theoretical Computer Science	Publication Date: Feb 12, 2020
Citations: 2	License type: cc-by

Affiliation: Utrecht University

Abstract

Based on analogies between algebraic curves and graphs, Baker and Norine introduced divisorial gonality, a graph parameter for multigraphs related to treewidth, multigraph algorithms and number theory. Various equivalent definitions of the gonality of an algebraic curve translate to different notions of gonality for graphs, called stable gonality and stable divisorial gonality.We consider so-called hyperelliptic graphs (multigraphs of gonality 2, in any meaning of graph gonality) and provide a safe and complete set of reduction rules for such multigraphs. This results in an algorithm to recognize hyperelliptic graphs in time O(m+nlog⁡n), where n is the number of vertices and m the number of edges of the multigraph. A corollary is that we can decide with the same runtime whether a two-edge-connected graph G admits an involution σ such that the quotient G/〈σ〉 is a tree.

Highlights

How complex is a graph G?1 If we start from the premise that the simplest connected graphs are trees, we can assign a complexity to G by stating how much it ‘deviates’, in a quantitative sense, from being a tree
We give polynomial time algorithms that recognize dgon(G) ≤ 2, sgon(G) ≤ 2 or sdgon(G) ≤ 2 for a graph G. (We do not consider gonality, which is defined without finiteness conditions.) To obtain our algorithms, we provide safe and complete sets of reduction rules
In [3], hyperelliptic graphs G are defined by dgon(G) = 2, and it is proven that, for twoconnected graphs G, this is equivalent to the existence of a harmonic morphism of degree two from G to a tree

Summary

Introduction

How complex is a graph G?1 If we start from the premise that the simplest connected graphs are trees, we can assign a complexity to G by stating how much it ‘deviates’, in a quantitative sense, from being a tree. The gonality of a curve is related to the stable gonality of a certain graph related to the equation by reduction modulo prime numbers, and our results are relevant in the theory of algorithms for diophantine equations [7]. Stable gonality sgon(G) (a priori defined using universal quantifiers over three infinite sets) is decidable [14], and computing divisorial gonality dgon(G) is NP-hard [13]. It follows from [10, §5] that computing divisorial gonality is in XP. (b) Intersection dual graph with finite harmonic morphism of degree p

Number theoretical applications

Preliminaries

Reduction rules for divisorial gonality

Reduction rules for stable gonality

Reduction rules for stable divisorial gonality

Algorithms

Conclusion