A note on labeling schemes for graph connectivity

Let G=(V,E) be an undirected graph and let S⊆V. The S-connectivity λGS(u,v) of u,v ε V is the maximum number of uv-paths that no two of them have an edge or a node in S{u,v} in common. Edge-connectivity is the case S=ø and node-connectivity is the case S=V. Given an integer k and a subset T⊆V of terminals, we consider the problem of assigning small "labels" (binary strings) to the terminals, such that given the labels of two terminals u,v ε T, one can decide whether λGS(u,v)≥k (k-partial labeling scheme) or to return min{λGS(u,v),k} (k-full labeling scheme). We observe that the best known labeling schemes for edge-connectivity (the case S=ø) extend to element-connectivity (the case S⊆VT), and use it to obtain a simple k-full labeling scheme for node-connectivity (the case S=V). If q distinct queries are expected, our k-full scheme has max-label size O(k^log2|T|logq), with success probability 1-1q for all queries. We also give a deterministic k-full labeling scheme with max-label size O(k ^log3|T|). Recently, Hsu and Lu (2009) [6] gave an optimal O(klog|T|) labeling scheme for the k-partial case. This implies an O(^k2log|T|) labeling scheme for the k-full case. Our deterministic k-full labeling scheme is better for k=Ω(^log2|T|).