Rainbow domination on trees Original Research Article

This paper studies a variation of domination in graphs called rainbow domination. For a positive integer image, a image-rainbow dominating function of a graph image is a function image from image to the set of all subsets of image such that for any vertex image with image we have image. The image-rainbow domination is the same as the ordinary domination. The image-rainbow domination problem is to determine the image-rainbow domination number image of a graph image, that is the minimum value of image where image runs over all image-rainbow dominating functions of image. In this paper, we prove that the image-rainbow domination problem is NP-complete even when restricted to chordal graphs or bipartite graphs. We then give a linear-time algorithm for the image-rainbow domination problem on trees. For a given tree image, we also determine the smallest image such that image.