Recognizing hyperelliptic graphs in polynomial time

Jelco M. Bodewes, Hans L. Bodlaender, Gunther Cornelissen, Marieke van der Wegen

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.

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