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

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 A, we show that the space requirement of our data structure is at least (16 n log2 A - 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 (n log2 A + 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 (n log2 n + O (n))-bit upper bound when A = O (nE), for any 0 < c < 1. Next, we consider the interval graphs with bounded chromatic number x, and design a ((x - 1)n + o(xn))-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 x = o(logn). Finally, we provide parameterized compact data structures for circular-arc graphs as well with bounded degree or bounded chromatic number condition.