Succinct Data Structures for Bounded Degree/Chromatic Number Interval Graphs

An interval graph is the intersection graph of intervals on the real line. We consider the problem of constructing space efficient data structures for two subclasses of interval graphs: those with maximum degree σ
1
and those with chromatic number at most σ
2
.We show that both bounded degree and bounded chromatic number interval graphs have a tight lower bound of n lg σ
i
− o(n lg σ
i
) bits (i = 1, 2). This improves the lower bound of Chakraborty and Jo from $\frac{1}{6}n\lg {\sigma _i} - O(n)$. For bounded chromatic number interval graphs, we give the first succinct data structure occupying n lg σ
2
+ O(n) bits that supports navigational operations and distance queries in O(σ
2
lgn) time. To match Chakraborty and Jo’s time complexity of O(lg lg σ
2
), which uses (σ
2
− 1)n+O(n) bits, we use 2nlgσ
2
+O(n) bits instead.