Compact representation of interval graphs and circular-arc graphs of bounded degree and chromatic number

Abstract

In this paper we initiate the study of designing parameterized compact data structures for an interval graph G with n vertices. First, we show that when the maximum degree of G is bounded by Δ, we show that the space requirement of our data structure is at least (16nlog2⁡Δ−O(n))-bit, which is one of the main contributions of this work. Next, by straightforward modifications of the result of Acan et al. (2021) [5], we obtain an (nlog2⁡Δ+O(n))-bit data structure supporting the standard navigational queries i.e., degree, adjacency, and neighborhood optimally. Hence, we provide a compact representation of interval graphs with bounded degree for the first time in literature. Note that this upper bound result takes less space than their (nlog2⁡n+O(n))-bit upper bound when Δ=O(nϵ), for any 0<ϵ<1. Next, we consider the interval graphs with bounded chromatic number ▪, and design a ▪-bit data structure with efficient query times. Unlike the previous upper bound, this data structure is completely new and doesn't follow from the result of Acan et al. (2021) [5]. Moreover, this takes less space than their data structure when ▪. Finally, we provide parameterized compact data structures for circular-arc graphs as well with bounded degree or bounded chromatic number condition.