Stable Divisorial Gonality is in NP

H.L. Bodlaender; Marieke van der Wegen; Tom van der Zanden

doi:10.1007/978-3-030-10801-4_8

Stable Divisorial Gonality is in NP

H.L. Bodlaender, Marieke van der Wegen^*, Tom van der Zanden

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Conference article in proceeding › Academic › peer-review

17 Downloads (Pure)

Abstract

Divisorial gonality and stable divisorial gonality are graph parameters, which have an origin in algebraic geometry. Divisorial gonality of a connected graph G can be defined with help of a chip firing game on G. The stable divisorial gonality of G is the minimum divisorial gonality over all subdivisions of edges of G.

In this paper we prove that deciding whether a given connected graph has stable divisorial gonality at most a given integer k belongs to the class NP. Combined with the result that (stable) divisorial gonality is NP-hard by Gijswijt, we obtain that stable divisorial gonality is NP-complete. The proof consists of a partial certificate that can be verified by solving an Integer Linear Programming instance. As a corollary, we have that the number of subdivisions needed for minimum stable divisorial gonality of a graph with n vertices is bounded by 2p(n) for a polynomial p.

Original language	English
Title of host publication	THEORY AND PRACTICE OF COMPUTER SCIENCE, SOFSEM 2019
Subtitle of host publication	45th International Conference on Current Trends in Theory and Practice of Computer Science, Nový Smokovec, Slovakia, January 27-30, 2019, Proceedings
Editors	Barbara Catania, Rastislav Královic, Jerzy Nawrocki, Giovanni Pighizzini
Publisher	Springer, Cham
Pages	81-93
Number of pages	13
ISBN (Electronic)	978-3-030-10801-4
ISBN (Print)	978-3-030-10800-7
DOIs	https://doi.org/10.1007/978-3-030-10801-4_8
Publication status	Published - 2019

Publication series

Series	Lecture Notes in Computer Science
Volume	11376
ISSN	0302-9743

Keywords

CURVES

Access to Document

10.1007/978-3-030-10801-4_8Licence: Unspecified

Full TextFinal published version, 292 KBLicence: Taverne

Cite this

Bodlaender, H. L., van der Wegen, M., & van der Zanden, T. (2019). Stable Divisorial Gonality is in NP. In B. Catania, R. Královic, J. Nawrocki, & G. Pighizzini (Eds.), THEORY AND PRACTICE OF COMPUTER SCIENCE, SOFSEM 2019: 45th International Conference on Current Trends in Theory and Practice of Computer Science, Nový Smokovec, Slovakia, January 27-30, 2019, Proceedings (pp. 81-93). Springer, Cham. https://doi.org/10.1007/978-3-030-10801-4_8

Bodlaender, H.L. ; van der Wegen, Marieke ; van der Zanden, Tom. / Stable Divisorial Gonality is in NP. THEORY AND PRACTICE OF COMPUTER SCIENCE, SOFSEM 2019: 45th International Conference on Current Trends in Theory and Practice of Computer Science, Nový Smokovec, Slovakia, January 27-30, 2019, Proceedings. editor / Barbara Catania ; Rastislav Královic ; Jerzy Nawrocki ; Giovanni Pighizzini. Springer, Cham, 2019. pp. 81-93 (Lecture Notes in Computer Science, Vol. 11376).

@inproceedings{4fcc22861a844a1f97d1f83116e09e7a,

title = "Stable Divisorial Gonality is in NP",

abstract = "Divisorial gonality and stable divisorial gonality are graph parameters, which have an origin in algebraic geometry. Divisorial gonality of a connected graph G can be defined with help of a chip firing game on G. The stable divisorial gonality of G is the minimum divisorial gonality over all subdivisions of edges of G.In this paper we prove that deciding whether a given connected graph has stable divisorial gonality at most a given integer k belongs to the class NP. Combined with the result that (stable) divisorial gonality is NP-hard by Gijswijt, we obtain that stable divisorial gonality is NP-complete. The proof consists of a partial certificate that can be verified by solving an Integer Linear Programming instance. As a corollary, we have that the number of subdivisions needed for minimum stable divisorial gonality of a graph with n vertices is bounded by 2p(n) for a polynomial p.",

keywords = "CURVES",

author = "H.L. Bodlaender and {van der Wegen}, Marieke and {van der Zanden}, Tom",

note = "data source: no data used",

year = "2019",

doi = "10.1007/978-3-030-10801-4_8",

language = "English",

isbn = "978-3-030-10800-7",

series = "Lecture Notes in Computer Science",

publisher = "Springer, Cham",

pages = "81--93",

editor = "Barbara Catania and Rastislav Kr{\'a}lovic and Jerzy Nawrocki and Giovanni Pighizzini",

booktitle = "THEORY AND PRACTICE OF COMPUTER SCIENCE, SOFSEM 2019",

address = "Switzerland",

}

Bodlaender, HL, van der Wegen, M & van der Zanden, T 2019, Stable Divisorial Gonality is in NP. in B Catania, R Královic, J Nawrocki & G Pighizzini (eds), THEORY AND PRACTICE OF COMPUTER SCIENCE, SOFSEM 2019: 45th International Conference on Current Trends in Theory and Practice of Computer Science, Nový Smokovec, Slovakia, January 27-30, 2019, Proceedings. Springer, Cham, Lecture Notes in Computer Science, vol. 11376, pp. 81-93. https://doi.org/10.1007/978-3-030-10801-4_8

Stable Divisorial Gonality is in NP. / Bodlaender, H.L.; van der Wegen, Marieke; van der Zanden, Tom.
THEORY AND PRACTICE OF COMPUTER SCIENCE, SOFSEM 2019: 45th International Conference on Current Trends in Theory and Practice of Computer Science, Nový Smokovec, Slovakia, January 27-30, 2019, Proceedings. ed. / Barbara Catania; Rastislav Královic; Jerzy Nawrocki; Giovanni Pighizzini. Springer, Cham, 2019. p. 81-93 (Lecture Notes in Computer Science, Vol. 11376).

Research output: Chapter in Book/Report/Conference proceeding › Conference article in proceeding › Academic › peer-review

TY - GEN

T1 - Stable Divisorial Gonality is in NP

AU - Bodlaender, H.L.

AU - van der Wegen, Marieke

AU - van der Zanden, Tom

N1 - data source: no data used

PY - 2019

Y1 - 2019

N2 - Divisorial gonality and stable divisorial gonality are graph parameters, which have an origin in algebraic geometry. Divisorial gonality of a connected graph G can be defined with help of a chip firing game on G. The stable divisorial gonality of G is the minimum divisorial gonality over all subdivisions of edges of G.In this paper we prove that deciding whether a given connected graph has stable divisorial gonality at most a given integer k belongs to the class NP. Combined with the result that (stable) divisorial gonality is NP-hard by Gijswijt, we obtain that stable divisorial gonality is NP-complete. The proof consists of a partial certificate that can be verified by solving an Integer Linear Programming instance. As a corollary, we have that the number of subdivisions needed for minimum stable divisorial gonality of a graph with n vertices is bounded by 2p(n) for a polynomial p.

AB - Divisorial gonality and stable divisorial gonality are graph parameters, which have an origin in algebraic geometry. Divisorial gonality of a connected graph G can be defined with help of a chip firing game on G. The stable divisorial gonality of G is the minimum divisorial gonality over all subdivisions of edges of G.In this paper we prove that deciding whether a given connected graph has stable divisorial gonality at most a given integer k belongs to the class NP. Combined with the result that (stable) divisorial gonality is NP-hard by Gijswijt, we obtain that stable divisorial gonality is NP-complete. The proof consists of a partial certificate that can be verified by solving an Integer Linear Programming instance. As a corollary, we have that the number of subdivisions needed for minimum stable divisorial gonality of a graph with n vertices is bounded by 2p(n) for a polynomial p.

KW - CURVES

U2 - 10.1007/978-3-030-10801-4_8

DO - 10.1007/978-3-030-10801-4_8

M3 - Conference article in proceeding

SN - 978-3-030-10800-7

T3 - Lecture Notes in Computer Science

SP - 81

EP - 93

BT - THEORY AND PRACTICE OF COMPUTER SCIENCE, SOFSEM 2019

A2 - Catania, Barbara

A2 - Královic, Rastislav

A2 - Nawrocki, Jerzy

A2 - Pighizzini, Giovanni

PB - Springer, Cham

ER -

Bodlaender HL, van der Wegen M, van der Zanden T. Stable Divisorial Gonality is in NP. In Catania B, Královic R, Nawrocki J, Pighizzini G, editors, THEORY AND PRACTICE OF COMPUTER SCIENCE, SOFSEM 2019: 45th International Conference on Current Trends in Theory and Practice of Computer Science, Nový Smokovec, Slovakia, January 27-30, 2019, Proceedings. Springer, Cham. 2019. p. 81-93. (Lecture Notes in Computer Science, Vol. 11376). doi: 10.1007/978-3-030-10801-4_8