On the global offensive alliance number of a graph
release_npkvr57tmrfkzlxxyxq2zv2ecq
by
J. M. Sigarreta, J. A. Rodriguez
2006
Abstract
An offensive alliance in a graph Γ=(V,E) is a set of vertices
S⊂ V where for every vertex v in its boundary it holds that the
majority of vertices in v's closed neighborhood are in S. In the case of
strong offensive alliance, strict majority is required. An alliance S is
called global if it affects every vertex in V S, that is, S is a
dominating set of Γ. The offensive alliance number a_o(Γ)
(respectively, strong offensive alliance number a_ô(Γ)) is the
minimum cardinality of an offensive (respectively, strong offensive) alliance
in Γ. The global offensive alliance number γ_o(Γ) and the
global strong offensive alliance number γ_ô(Γ) are defined
similarly. Clearly, a_o(Γ)<γ_o(Γ) and
a_ô(Γ)<γ_ô(Γ). It was shown in [Discuss.
Math. Graph Theory, 24 (2004), no. 2, 263-275] that a_o(Γ)<2n/3 and a_ô(Γ)<5n/6, where n denotes the
order of Γ. In this paper we obtain several tight bounds on
γ_o(Γ) and γ_ô(Γ) in terms of several
parameters of Γ. For instance, we show that 2m+n/3Δ+1<γ_o(Γ)<2n/3 and 2(m+n)/3Δ+2<γ_ô(Γ)<5n/6, where m denotes the size of
Γ and Δ its maximum degree (the last upper bound holds true for
all Γ with minimum degree greatest or equal to two).
In text/plain
format
Archived Files and Locations
application/pdf 187.3 kB
file_ozpwi7deizdyjmfhi25afkjh4u
|
archive.org (archive) |
math/0602432v1
access all versions, variants, and formats of this works (eg, pre-prints)