On links minimizing the tunnel number

Girão, Darlan; Nogueira, João Miguel; Salgueiro, António

doi:10.1142/s0218216519400145

Cited by 2 publications

(3 citation statements)

References 8 publications

(9 reference statements)

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“…The result of this subsection have appeared, in a more general setting, in [13]. We include it here for completeness.…”

Section: Relationship Between Hull Number and Tunnel Numbermentioning

confidence: 90%

“…What is the smallest unknotting tunnel system consisting of vertical arcs only? One of the results in [13] shows that this is at most the hull number associated to the interval function I fc of the graph G D(K) . We conjecture that the tunnel number is also bounded above by hn cc .…”

Section: Tunnel Numbermentioning

confidence: 97%

“…This lower bound can be used to determine the tunnel number of a link in case one finds an unknotting tunnel system with that cardinality, as in [12]. Also in [13] the authors determine the tunnel number of a class of links by exploring the following upper bound: given a diagram D(K) of K, consider vertical arcs added at certain crossings (see figure 2 below). What is the smallest unknotting tunnel system consisting of vertical arcs only?…”

Section: Tunnel Numbermentioning

confidence: 99%

See 2 more Smart Citations

Cycle convexity and the tunnel number of links

Araújo¹,

Campos²,

Girão³

et al. 2020

Preprint

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In this work, we introduce a new graph convexity, that we call Cycle Convexity, motivated by related notions in Knot Theory.For a graph G = (V, E), define the interval function in the Cycle Convexity asThe hull number of G in the cycle convexity, denoted by hn cc (G), is the cardinality of a smallest hull set of G.We first present the motivation for introducing such convexity and the study of its related hull number. Then, we prove that: the hull number of a 4-regular planar graph is at most half of its vertices; computing the hull number of a planar graph is an NP-complete problem; computing the hull humber of chordal graphs, P 4 -sparse graphs and grids can be done in polynomial time.

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“…The result of this subsection have appeared, in a more general setting, in [13]. We include it here for completeness.…”

Section: Relationship Between Hull Number and Tunnel Numbermentioning

confidence: 90%

Section: Tunnel Numbermentioning

confidence: 97%