Recognizing hyperelliptic graphs in polynomial time

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. We consider so-called hyperelliptic graphs (multigraphs of gonality 2) and provide a safe and complete set of reduction rules for such multigraphs, showing that we can recognize hyperelliptic graphs in time O(n log n+ m), 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.