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Mitre C. Dourado; Dieter Rautenbach; Vinícius Fernandes dos Santos; Philipp M. Schäfer; Jayme L. Szwarcfiter; Alexandre Toman

doi:10.1016/j.disc.2012.05.002

An upper bound on the P3-Radon number

Discrete Mathematics ◽

10.1016/j.disc.2012.05.002 ◽

2012 ◽

Vol 312 (16) ◽

pp. 2433-2437 ◽

Cited By ~ 16

Author(s):

Mitre C. Dourado ◽

Dieter Rautenbach ◽

Vinícius Fernandes dos Santos ◽

Philipp M. Schäfer ◽

Jayme L. Szwarcfiter ◽

...

Keyword(s):

Upper Bound ◽

Radon Number

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Complexity aspects of the computation of the rank of a graph

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2075 ◽

2014 ◽

Vol Vol. 16 no. 2 (PRIMA 2013) ◽

Author(s):

Igor Ramos ◽

Vinícius F. Santos ◽

Jayme L. Szwarcfiter

Keyword(s):

Convex Hull ◽

Upper Bound ◽

Convex Set ◽

Small Diameter ◽

Independent Set ◽

Special Issue ◽

Undirected Graphs ◽

Radon Number ◽

International Audience ◽

Np Complete

Special issue PRIMA 2013 International audience We consider the P₃-convexity on simple undirected graphs, in which a set of vertices S is convex if no vertex outside S has two or more neighbors in S. The convex hull H(S) of a set S is the smallest convex set containing S as a subset. A set S is a convexly independent set if v \not ∈ H(S\setminus \v\) for all v in S. The rank \rk(G) of a graph is the size of the largest convexly independent set. In this paper we consider the complexity of determining \rk(G). We show that the problem is NP-complete even for split or bipartite graphs with small diameter. We also show how to determine \rk(G) in polynomial time for the well structured classes of graphs of trees and threshold graphs. Finally, we give a tight upper bound for \rk(G), which in turn gives a tight upper bound for the Radon number as byproduct, which is the same obtained before by Henning, Rautenbach and Schäfer. Additionally, we briefly show that the problem is NP-complete also in the monophonic convexity.

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