On defensive alliances and strong global offensive alliances

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 γô(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 γô(G) ≤ 3n(G)/4, which improves previous results by Favaron et al. and Sigarreta and Rodŕıguez. Finally, we prove γô(G) ≤ ( 1 2 + (1 + o(δ(G))) ln(δ(G)+1) δ(G)+1 ) n(G).