Powerful Alliances in Graphs
For a graph G = (V, E), a non-empty set S ⊆ V is a defensive alliance if for every vertex v in S, v has at most one more neighbor in V - S than it has in S, and S is an offensive alliance if for every v ∈ V - S that has a neighbor in S, v has more neighbors in S than in V - S. A powerful alliance is both defensive and offensive. We initiate the study of powerful alliances in graphs.
Brigham, Robert C.; Dutton, Ronald D.; Haynes, Teresa W.; and Hedetniemi, Stephen T.. 2009. Powerful Alliances in Graphs. Discrete Mathematics. Vol.309(8). 2140-2147. https://doi.org/10.1016/j.disc.2006.10.026 ISSN: 0012-365X