Complexity aspects of the computation of the rank of a graph

Igor da Fonseca Ramos (1, 2, 3) , Vinícius F. Dos Santos (4, 5) , Jayme L. Szwarcfiter (1, 6, 2, 3)

1 DCC - Departamento de Ciência Da Computação [Rio de Janeiro]
2 PESC - Programa de Engenharia de Sistemas e Computação
3 COPPE-UFRJ - Instituto Alberto Luiz Coimbra de Pós-Graduação e Pesquisa de Engenharia
4 Departamento de Matematica Aplicada (Rio de Janeiro)
5 IME - Instituto de Matemática e Estatística (Rio do Janeiro)
6 NCE - Nucleo de Computação Eletrônica

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.

Mots clés: Algorithm Design; Graph Theory; Graph Convexity; Computational Complexity

Domaines: Mathématique discrète [cs.DM]

Igor da Fonseca Ramos, Vinícius F. Dos Santos, Jayme L. Szwarcfiter. Complexity aspects of the computation of the rank of a graph. Discrete Mathematics and Theoretical Computer Science, 2014, Vol. 16 no. 2 (2), pp.73--86. ⟨10.46298/dmtcs.2075⟩. ⟨hal-01185615⟩