Ramos, Igor, and Santos, Vinícius F., and Szwarcfiter, Jayme L. - Complexity aspects of the computation of the rank of a graph

dmtcs:2075 - Discrete Mathematics & Theoretical Computer Science, September 1, 2014, Vol. 16 no. 2
Complexity aspects of the computation of the rank of a graph

Authors: Ramos, Igor, and Santos, Vinícius F., and Szwarcfiter, Jayme L.

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.


Source : oai:HAL:hal-01185615v1
Volume: Vol. 16 no. 2
Section: PRIMA 2013
Published on: September 1, 2014
Submitted on: December 2, 2013
Keywords: Algorithm Design,Graph Theory,Graph Convexity,Computational Complexity,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]


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