Complexity results related to monophonic convexity

Dourado, Mitre C.; Protti, Fábio; Szwarcfiter, Jayme Luiz

doi:10.1016/j.dam.2009.11.016

Cited by 85 publications

(35 citation statements)

References 16 publications

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“…Several well known graph convexities C are defined using some set P of paths of the underlying graph G. In this case, a subset S of vertices of G is convex, that is, belongs to C, if for every path P in P whose endvertices belong to S also every vertex of P belongs to S. When P is the set of all shortest paths in G, this leads to the geodetic convexity [7,15,16,22,24]. The monophonic convexity is defined by considering as P the set of all induced paths of G [17,20,23]. The set of all paths of G leads to the all path convexity [12].…”

Section: The Carathéodory Number Cth(g) Is the Smallest Integer C Sucmentioning

confidence: 99%

Inapproximability Results for Graph Convexity Parameters

Coelho

Dourado

Sampaio

2014

Approximation and Online Algorithms

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Abstract. In this paper, we prove several inapproximability results on the P3-convexity and the geodetic convexity on graphs. We prove that determining the P3-hull number and the geodetic hull number are APXhard problems. We prove that the Carathéodory number, the Radon number and the convexity number of both convexities are O(n 1−ε )-inapproximable in polynomial time for every ε > 0, unless P=NP. We also prove that the interval numbers of both convexities are W [2]-hard and O(log n)-inapproximable in polynomial time, unless P=NP. Moreover, these results hold for bipartite graphs in the P3-convexity.

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Section: The Carathéodory Number Cth(g) Is the Smallest Integer C Sucmentioning

confidence: 99%

Inapproximability Results for Graph Convexity Parameters

Coelho

Dourado

Sampaio

2014

Approximation and Online Algorithms

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“…Further results involving computational complexity problems related to geodesic convexity in graphs can be found in [3,4,6,38,80,81,83,84,86,95,104,110,113,123,140,164].…”

Section: Theorem 74 ([104]) the Convexity Number Problem Is Np-compmentioning

confidence: 99%

Geodesic Convexity in Graphs

Pelayo¹

2013

SpringerBriefs in Mathematics

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SpringerBriefs in Mathematics showcases expositions in all areas of mathematics and applied mathematics. Manuscripts presenting new results or a single new result in a classical field, new field, or an emerging topic, applications, or bridges between new results and already published works, are encouraged. The series is intended for mathematicians and applied School of Agricultural Engineering of Barcelona

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“…Next to the geodetic convexity, further well-studied graphs convexities are the P 3 -convexity [5], the induced paths convexity, also known as the monophonic convexity [12,14,18], the all paths convexity [6], the triangle path convexity [7,8], and the convexity based on induced paths of order at least 4 [13]. Our contributions are as follows.…”

Section: Introductionmentioning

confidence: 99%