Succinct Encodings for Families of Interval Graphs

Succinct data structure for path graphs

“…In this paper, we add to this growing body of work by presenting a data structure for the class of n vertex graphs, introduced by Erdős and West in [21], with d−dimensional t−representation, denoted by G t,d . They proved an upper bound of (2nt+ 1 2 )d log n−(n− [16]. There seems to be no other result on G t,d after [21], to the best of our knowledge.…”

“…Now, we describe an alternative algorithm for neighbor queries when d = 1, using an additional O(tn) bits of space. In this specific case, we utilize the following lemma, which is derived from the work of Acan et al [16] for interval graphs. Also in this case, we use I v p to denote I v p,1 .…”

“…A recent paper by Chakraborty and Jo [5] improve the information theoretic lower bound for interval graphs as given by Acan et al [2] by bounding maximum degree. For bounded maximum degree ∆ in O(n ϵ ) where 1 < ϵ < 1, they give a lower bound of ( 16 n log ∆ − O(n))−bits and a (n log ∆ + O(n))−bit space-efficient data structure.…”

“…In recent years, succinct data structures for intersection graphs is getting lot of attention. A few of them are, Acan et al [2] for interval graphs, Acan et al [1] for families of intersection graphs on circle, Munro and Wu [19] for chordal graphs and Balakrishnan et al [4] for path graphs. Also, Markenzon et al [16] gives an efficient representation for chordal graphs.…”

“…Succinct data structures for intersection graphs, particularly interval graphs, and their generalizations have already appeared; see Golumbic [15] for more on interval graphs. For interval graphs, Acan et al in [16] (and later He et al [17]) give a (n log n + O(n))-bit succinct data structure that supports adjacency, neighbourhood and degree queries in constant time. Class of chordal graphs is a strict super-class of interval graphs and Munro and Wu in [18] have given a (n 2 /4 + o(n 2 ))-bit succinct data structure that supports the queries efficiently.…”