Discrete Applied Mathematics 141 (2004) 119 – 134
www.elsevier.com/locate/dam
On the complexity of the approximation of
nonplanarity parameters for cubic graphs
Luerbio Faria
a
, Celina M. Herrera de Figueiredo
b
,
Candido F.X. Mendonca
c
a
Faculdade de Formac
˜
ao de Professores, Universidade do Estado do, Rio de Janeiro, Brazil
b
Instituto de Matem
Ã
atica and COPPE, Universidade Federal do Rio de Janeiro, Brazil
c
Departamento de Inform
Ã
atica, Universidade Estadual de Maring
Ã
a, Brazil
Received 22 June 2001; received in revised form 29 May 2002; accepted 22 March 2003
Abstract
Let G =(V; E) be a simple graph. The
NON-PLANAR DELETION
problem consists in ÿnding a
smallest subset E
⊂ E such that H =(V; E\E
) is a planar graph. The
SPLITTING NUMBER
problem
consists in ÿnding the smallest integer k ¿ 0, such that a planar graph H can be deÿned from
G by k vertexsplitting operations. We establish the MaxSNP-hardness of
SPLITTING NUMBER
and
NON-PLANAR DELETION
problems for cubic graphs.
? 2003 Elsevier B.V. All rights reserved.
Keywords: Topological graph theory; Complexity classes; Computational diculty of problems; Splitting
number; Maximum planar subgraph
1. Introduction
Let G =(V; E) be a simple graph. The
NON-PLANAR DELETION
problem consists in
ÿnding a smallest subset E
⊂ E such that H =(V; E \ E
) is a planar graph. The
MAXIMUM PLANAR SUBGRAPH
problem consists in ÿnding a largest subset E
⊂ E such that
H =(V; E
) is a planar graph. Given u ∈ V (G), say that a graph H is obtained from G
by splitting vertex u if V (H )=(V (G) \{u}) ∪{u
1
;u
2
} and E(H )=(E(G) \{(u; x):
x ∈ N (u)}) ∪{(u
1
;x):x ∈ N
1
}∪{(u
2
;x):x ∈ N
2
}, where N (u), the neighborhood of
u in G, is partitioned into non-empty sets N
1
and N
2
. The
SPLITTING NUMBER
problem
Partially supported by CNPq, CAPES, FAPERJ, FINEP, Brazilian research agencies.
E-mail addresses: luerbio@cos.ufrj.br (L. Faria), celina@cos.ufrj.br (C.M. Herrera de Figueiredo),
xavier@din.uem.br (C.F.X. Mendonca).
0166-218X/$ - see front matter ? 2003 Elsevier B.V. All rights reserved.
doi:10.1016/S0166-218X(03)00370-6