On defensive alliances and strong global offensive alliances

Authors:
Mitre C. Dourado;Luerbio Faria;Miguel A. Pizaña;Dieter Rautenbach;Jayme L. Szwarcfiter
Affiliations:
-;-;-;-;-
Venue:
Discrete Applied Mathematics
Year:
2014

Citing 5
Cited 0

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Algorithms and complexity for alliances and weighted alliances of various types

Algorithms and complexity for alliances and weighted alliances of various types
On the global offensive alliance number of a graph

Discrete Applied Mathematics

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Abstract

We consider complexity issues and upper bounds for defensive alliances and strong global offensive alliances in graphs. We prove that it is NP-complete to decide for a given 6-regular graph G and a given integer a, whether G contains a defensive alliance of order at most a. Furthermore, we prove that determining the strong global offensive alliance number @c"o"@?(G) of a graph G is APX-hard for cubic graphs and NP-hard for chordal graphs. For a graph G of minimum degree at least 2, we prove @c"o"@?(G)@?3n(G)/4, which improves previous results by Favaron et al. and Sigarreta and Rodriguez. Finally, we prove @c"o"@?(G)@?(12+(1+o(@d(G)))ln(@d(G)+1)@d(G)+1)n(G).