Abstract :
In a graph image of order image and maximum degree image, a subset image of vertices is a image-independent set if the subgraph induced by image has maximum degree less or equal to image. The lower image-independence number image image is the minimum cardinality of a maximal image-independent set in image and the image-independence number image is the maximum cardinality of a image-independent set. We show that image for any graph and any image, and image if image is connected, that image for any tree, and that image for any connected bipartite graph with image support vertices. Moreover, we characterize the trees satisfying image, image, image or image.
