Boundary defensive image-alliances in graphs Original Research Article

We define a boundary defensive image-alliance in a graph as a set image of vertices with the property that every vertex in image has exactly image more neighbors in image than it has outside of image. In this paper we study mathematical properties of boundary defensive image-alliances. In particular, we obtain several bounds on the cardinality of every boundary defensive image-alliance. Moreover, we consider the case in which the vertex set of a graph can be partitioned into boundary alliances, showing that if a image-regular graph image of order image can be partitioned into two boundary defensive image-alliances image and image, then image and the algebraic connectivity of image is equal to image.