Simple labeling schemes for graph connectivity

Let G = (V,E) be an undirected graph and let S ⊆ V . The S-connectivity SG(u, v) of a pair of nodes u, v in G 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 = ∅ and node-connectivity is the case S = V . Given a graph G = (V,E), an integer k, a subset T ⊆ V of terminals, and S ⊆ V , we consider the problem of assigning small “labels” (binary strings) to the terminals, so that given the labels of two terminals u, v ∈ T , one can decide whether SG(u, v) ≥ k (k-partial labeling scheme) or to return min{ SG(u, v), k} (k-full labeling scheme). For edge-connectivity, there are known labeling schemes with max-label size O(log ∣T ∣) in the k-partial case and O(log ∣T ∣ ⋅ min{k, log ∣T ∣}) in the k-full case [3]. We observe that this result extends to S-connectivity when S ⊆ V ∖T – so called “element-connectivity”, and combine it with the recently discovered decomposition of Chuzhoy and Khanna [1] of node-connectivity problems into element-connectivity problems to obtain simple labeling schemes for node-connectivity. If q distinct queries are expected, our labeling schemes have max-label size O(k log ∣T ∣ log q) in the k-partial case and O(k log ∣T ∣ log q⋅min{k, log ∣T ∣}) in the k-full case, with success probability 1− 1 q for all queries. For a constant number of queries, this matches the lower bound (k log n) for the k-partial case of [3]. Consequently, we obtain deterministic labeling schemes with max-label size O(k log ∣T ∣) in the k-partial case and O(k log ∣T ∣ ⋅ min{k, log ∣T ∣}) in the k-full case. This improves the bounds of Korman [4] O(k log n) and O(k log n), respectively, for k = (log ∣T ∣).