Improved sparse covers for graphs excluding a fixed minor

Costas Busch, Ryan LaFortune, Srikanta Tirthapura

Abstract: We consider the construction of sparse covers for planar graphs and other graphs that exclude a fixed minor. We present an algorithm that gives a cover for the γ-neighborhood of each node. For planar graphs, the cover has radius no more than 24γ-8 and degree (maximum cluster overlaps) no more than 18. For every n node graph that excludes a fixed minor, we present an algorithm that yields a cover with radius no more than 4γ and degree O(log n). This is a significant improvement over previous results for planar graphs and for graphs excluding a fixed minor; in order to obtain clusters with radius of O(γ), it was required to have degree polynomial in n. Since sparse covers have many applications in distributed computing, including compact routing, distributed directories and synchronizers, our improved cover construction results in improved algorithms for all these problems, for the class of graphs that exclude a fixed minor.