Succinct navigational oracles for families of intersection graphs on a circle

Succinct data structure for path graphs

“…. , P i ki ∈ P ′ i , we index paths of G and H in the following ways: 1. in G paths are ordered {P 1 1 , . .…”

“…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.…”

“…1. Demonstrate an improved worst-case information theoretic lower bound for chordal graphs with bounded vertex leafage by using a simple and constructive counting technique motivated by the method of partial coloring as used by Acan et al [1]. 2.…”

“…Section 2 gives details of the compact data structures we have used in the construction of the succinct data structure of k−vertex leafage chordal graphs along with characterisation of chordal and path graphs. It also formalizes and explains the method of partial coloring, first used in Acan et al [1] to obtain the lower bound for ▶ Theorem 4. The graph G is a path graph if and only if there exists a clique tree T , such that for every v ∈ V (G), the set of maximal cliques containing v is a path in T such that two vertices are adjacent if the corresponding paths intersect.…”

“…pathep(P i[1]) (a 1 , b 1 ),P i 2 = (a 2 , b 2 ), l ← lca(a 1 , b 1 ), l ′ ← l 21 if lmost_child(parent(l)) = l then 22 v ← parent(getHPStartNode(l)) Add getNhb(v, l ′ , b 1 ) to N 27 l ← v 28 if child(1, parent(l)) = l then29 v ← parent(getHPStartNode(l)) 30 else 31 v ← parent(l) connecting the leaves. Further, as per Lemma 6 of [4], there are at most O(log n) light edges in any path of T .…”