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Recognizing hyperelliptic graphs in polynomial time
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.Comment: 33 pages, 8 figure
Computing graph gonality is hard
There are several notions of gonality for graphs. The divisorial gonality
dgon(G) of a graph G is the smallest degree of a divisor of positive rank in
the sense of Baker-Norine. The stable gonality sgon(G) of a graph G is the
minimum degree of a finite harmonic morphism from a refinement of G to a tree,
as defined by Cornelissen, Kato and Kool. We show that computing dgon(G) and
sgon(G) are NP-hard by a reduction from the maximum independent set problem and
the vertex cover problem, respectively. Both constructions show that computing
gonality is moreover APX-hard.Comment: The previous version only dealt with hardness of the divisorial
gonality. The current version also shows hardness of stable gonality and
discusses the relation between the two graph parameter
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