An Optimal Labeling for Node Connectivity

Given an n-node undirected simple graph G and a positive integer k, the k-connectivity labeling problem for G seeks short labels for the nodes of G such that whether any two nodes are k-connected in G can be determined merely by their labels. For k = 1, an optimal solution to the problem is to give each node in the same connected component of G a common log2 n -bit label, uniquely chosen for this connected component. For k ≥ 2, Katz, Katz, Korman, and Peleg gave the first nontrivial solution to the problem, requiring O(2 log n) bits per node. The best previously known solution, due to Korman, requires O(k log n) bits per node. We give the first asymptotically optimal solution to the problem, requiring only (2k− 1) log2 n bits per node, which matches a lower bound Ω(k log n) proved by Katz, Katz, Korman, and Peleg.