A note on labeling schemes for graph connectivity

Let G=(V,E)G=(V,E) be an undirected graph and let S⊆VS⊆V. The S  -connectivity λGS(u,v) of u,v∈Vu,v∈V is the maximum number of uv  -paths that no two of them have an edge or a node in S∖{u,v}S∖{u,v} in common. Edge-connectivity is the case S=∅S=∅ and node-connectivity is the case S=VS=V. Given an integer k   and a subset T⊆VT⊆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∈Tu,v∈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=∅S=∅) extend to element-connectivity (the case S⊆V∖TS⊆V∖T), and use it to obtain a simple k  -full labeling scheme for node-connectivity (the case S=VS=V). If q distinct queries are expected, our k  -full scheme has max-label size O(klog2|T|logq), with success probability 1−1q for all queries. We also give a deterministic k  -full labeling scheme with max-label size O(klog3|T|)O(klog3|T|). Recently, Hsu and Lu (2009) [6] gave an optimal O(klog|T|)O(klog|T|) labeling scheme for the k  -partial case. This implies an O(k2log|T|)O(k2log|T|) labeling scheme for the k-full case. Our deterministic k  -full labeling scheme is better for k=Ω(log2|T|)k=Ω(log2|T|).

► Let κuvκuv denote the uv  -node-connectivity in a graph G=(V,E)G=(V,E) and let k   be an integer.
► We consider assigning small node labels so that any u,v∈Vu,v∈V can compute min{κuv,k}min{κuv,k}.
► Our max-label size is O(klog3|V|)O(klog3|V|).
► For k=Ω(log2|V|)k=Ω(log2|V|) this improves the known bound O(k2log|V|)O(k2log|V|).