The convexity of induced paths of order three and applications: Complexity aspects

Araújo, Rafael T.; Sampaio, Rudini Menezes; Szwarcfiter, Jayme Luiz

doi:10.1016/j.dam.2017.11.007

Cited by 10 publications

(7 citation statements)

References 17 publications

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“…It is important to emphasize that there are several similar functions defined over the vertices of a graph in the literature, leading to several graph convexities, e.g. : geodetic [5], P 3 [6], P * 3 [7], monophonic [8]. The graph convexities we should emphasize are the P 3 and P * 3 .…”

Section: Graph Convexitiesmentioning

confidence: 99%

“…the authors prove that computing the hull number is NP-complete in general, and they give polynomial results for chordal graphs and cographs. In [7], the authors prove that computing the hull number in the P 3 -convexity remains NP-hard even when restricted to subgraphs of the grid. It implies that it is NP-hard for bipartite planar graphs with bounded maximum degree.…”

mentioning

confidence: 99%

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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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Section: Graph Convexitiesmentioning

confidence: 99%

mentioning

confidence: 99%

Cycle convexity and the tunnel number of links

Araújo¹,

Campos²,

Girão³

et al. 2020

Preprint

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show abstract

“…The classical convexity invariants Carathéodory, Helly, and Radon numbers have already been determined for the triangle path [5] and the monophonic [12,18] All these six problems have already been considered in the geodetic [1,8,14,16], monophonic [7,10], and P 3 [4,2] convexities.…”

Section: Introductionmentioning

confidence: 99%

“…All these six problems have already been considered in the geodetic [1,8,14,16], monophonic [7,10], and P 3 [4,2] convexities.…”

mentioning

confidence: 99%

Complexity aspects of the triangle path convexity

Dourado

Sampaio

2016

Discrete Applied Mathematics

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A path P = v1, . . . , vt is a triangle path (respectively, monophonic path) of G if no edges exist joining vertices vi and vj of P such that |j − i| > 2; (respectively, |j − i| > 1). A set of vertices S is convex in the triangle path convexity (respectively, monophonic convexity) of G if the vertices of every triangle path (respectively, monophonic path) joining two vertices of S are in S. The cardinality of a maximum proper convex set of G is the convexity number of G and the cardinality of a minimum set of vertices whose convex hull is V (G) is the hull number of G. Our main results are polynomial time algorithms for determining the convexity number and the hull number of a graph in the triangle path convexity.

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“…Morris [29] and Riedl [32] studied the maximum size of minimal percolating sets on the square grid [n] 2 and the hypercube {0, 1} d , respectively, answering a question posed by Bollobás. However, the problem of finding the smallest percolating set is NP-hard even on subgraphs of the square grid [2] and it is APX-hard even for bipartite graphs with maximum degree four [18]. Moreover, it is hard [17] to approximate within a ratio O(2 log 1−ε n ), for any ε > 0, unless N P ⊆ DT IM E(n polylog(n) ).…”

Section: Introductionmentioning

confidence: 99%

The Maximum Time of 2-neighbour Bootstrap Percolation in Grid Graphs and Parametrized Results

Marcilon

Sampaio

2016

Graph-Theoretic Concepts in Computer Science

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In 2-neighborhood bootstrap percolation on a graph G, an infection spreads according to the following deterministic rule: infected vertices of G remain infected forever and in consecutive rounds healthy vertices with at least two already infected neighbors become infected. Percolation occurs if eventually every vertex is infected. The maximum time t(G) is the maximum number of rounds needed to eventually infect the entire vertex set. In 2013, it was proved by Benevides et al [10] that t(G) is NP-hard for planar graphs and that deciding whether t(G) ≥ k is polynomial time solvable for k ≤ 2, but is NP-complete for k ≥ 4. They left two open problems about the complexity for k = 3 and for planar bipartite graphs. In 2014, we solved the first problem [28]. In this paper, we solve the second one by proving that t(G) is NP-complete even in grid graphs with maximum degree 3. We also prove that t(G) is polynomial time solvable for solid grid graphs with maximum degree 3. Moreover, we prove that the percolation time problem is W[1]-hard on the treewidth of the graph, but it is fixed parameter tractable with parameters treewidth+k and maxdegree+k.

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The convexity of induced paths of order three and applications: Complexity aspects

Cited by 10 publications

References 17 publications

Cycle convexity and the tunnel number of links

Cycle convexity and the tunnel number of links

Complexity aspects of the triangle path convexity

The Maximum Time of 2-neighbour Bootstrap Percolation in Grid Graphs and Parametrized Results

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