Recognizing hyperelliptic graphs in polynomial time

Abstract

Recently, a new set of multigraph parameters was defined, called "gonalities". Gonality bears some similarity to treewidth, and is a relevant graph parameter for problems in number theory and multigraph algorithms. Multigraphs of gonality 1 are trees. We consider so-called "hyperelliptic graphs" (multigraphs of gonality 2) and provide a safe and complete sets of reduction rules for such multigraphs, showing that for three of the flavors of gonality, we can recognize hyperelliptic graphs in O(n log n + m) time, where n is the number of vertices and m the number of edges of the multigraph. 1 To simplify terminology, we will use the word "graph" for a finite multigraph (i.e. allowing finitely many edges between the same vertices), and use the terminology "simple graph" if there is at most one edge between any pair of vertices.

Problems Hard for Treewidth but Easy for Stable Gonality

“…Is stable gonality fixed parameter tractable? Can multigraphs of fixed stable gonality be recognized efficiently (this holds for treewidth; for sgon = 2 this can be done in quasilinear time [7])? Given the stable gonality of a graph, can a refinement and morphism of that degree to a tree be constructed in reasonable time (the analogous problem for treewidth can be done in linear time)?…”

“…One replaces tree decompositions of a graph G by graph morphisms from G to trees, and the "width" of the decomposition by the "degree" of the morphism, where lower degree maps correspond to less complex graphs. For example, connected graphs of stable gonality 1 are trees [7,Example 2.13], those of stable gonality 2 are so-called hyperelliptic graphs, i.e., graphs that admit, after refinement, a graph automorphism of order two such that the quotient graph is a tree (decidable in quasilinear time [7,Thm. 6.1]).…”

Stable Divisorial Gonality is in NP

“…It is known that treewidth is a lower bound for stable divisorial gonality [7]. The stable divisorial gonality of a graph is at most the divisorial gonality, but this inequality can be strict, see for example [8,Figure 2].…”

“…We finish this introduction by giving an overview of the few previously known results on the algorithmic complexity of (stable) divisorial gonality. Bodewes et al [8] showed that deciding whether a graph has (stable) divisorial gonality at most 2 can be done in O(n log n + m) time. From [10] and [11], it follows that divisorial gonality belongs to the class XP, i.e.…”

Complexity of Graph Problems: Gonality, Colouring and Scheduling

“…Stable gonality has an application in the theory of Diophantine equations. A classical result about 'uniform boundedness' of Frey [50], using heavily This chapter is based on the introductions and preliminaries of [51,54,21,17].…”