Stability of Alliance Number in Graphs Grady D. Bullington Linda Eroh Steven J. Winters Department of Mathematics University of Wisconsin Oshkosh Oshkosh, WI 54901 USA June 29, 2005 Abstract In [1], Kristiansen, Hedetniemi and Hedetniemi introduce the con- cept of defensive alliance, defensive alliance number, and similar def- initions for many other types of alliance (e.g., strong defensive, of- fensive). In our work, we ask which graphs retain the same defensive alliance number (resp., strong defensive alliance number) when a single vertex or edge is deleted and determine the answer for several classes of graphs. Characterizations are also given for graphs which are stable in the above manner and possess certain low defensive and strong defensive alliance numbers. Introduction. For any vertex v of a graph G, the open neighborhood of v is N (v)= {u ∈ V (G): uv ∈ E(G)}, and the closed neighborhood of v is N [v]= N (v) ∪{v}. By [1], a non-empty set of vertices S ⊆ V of a graph G is called a defensive alliance if for every v ∈ S, |N [v] ∩S|≥|N (v) ∩(V −S)|. A defensive alliance is said to be strong if this inequality is strict. Further, the defensive alliance number a(G) is defined to be the smallest order of any defensive alliance of a graph G. Similarly, the strong defensive alliance number ˆ a(G) is the smallest order of any strong defensive alliance of G. Any alliance of G with cardinality a(G) is called an a-set of G, and any strong alliance of cardinality ˆ a(G) is called an ˆ a-set of G. Definition 1. We say that a nontrivial graph G is edge (resp., vertex ) a-stable if the deletion of any edge (resp., vertex) of G yields a graph with