Recognizing hyperelliptic graphs in polynomial time

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

doi:10.48550/arxiv.1706.05670

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“…has gonality 2: there is an algorithm in Magma for this problem on curves [11], but a rigorous analysis of its complexity has not been carried out. However, there are algorithms that decide this for both notions of gonality for graphs in quasilinear time [10].…”

Section: Computational Complexitymentioning

confidence: 99%

Computing graph gonality is hard

Gijswijt¹,

Smit²,

Wegen³

2015

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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.

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Section: Computational Complexitymentioning

confidence: 99%