Compact representation of graphs with bounded bandwidth or treedepth

Abstract

We study the problem of compact representation of graphs that have small bandwidth as well as graphs that have small treedepth. We present navigation oracles that support degree and adjacency queries in constant time and neighborhood queries in constant time per neighbor. For graphs of size n and bandwidth k, our oracle takes (k+⌈log⁡2k⌉)n+o(kn) bits. We also show that (k−5k−4)n−O(k2) bits are required to encode such graphs. For graphs of size n and treedepth k, we present an oracle that takes (k+⌈log⁡k⌉+2)n+o(kn) bits and complement it with lower bounds that show our oracle is compact for specific ranges of k. Our navigation oracles for both treedepth and treedepth parameters can be augmented, with additional n+o(n) bits, to support connectivity queries in constant time.