Compact Representation of Graphs of Small Clique-Width

• Published in: Algorithmica Vol. 80; no. 7; pp. 2106 - 2131
• Main Author: Kamali, Shahin
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-07-2018; Springer Nature B.V
• Subjects: Algorithm Analysis and Problem Complexity; Algorithms; Computer Science; Computer Systems Organization and Communication Networks; Data structures; Data Structures and Information Theory; Graphical representations; Graphs; Mathematics of Computing; Navigation; Preprocessing; Queries; Special Issue on Compact Data Structures; Theory of Computation; Compact representation; Navigation oracles; Clique-width
• DOI: 10.1007/s00453-017-0365-6
• EISSN: 1432-0541
• ISSN: 0178-4617

The notion of clique-width for graphs offers many research topics and has received considerable attention in the past decade. A graph has clique-width
k
if it can be represented as an algebraic expression on
k
labels associated with its vertices. Many computationally hard problems can be solved in polynomial time for graphs of bounded clique-width. Interestingly also, many graph families have bounded clique-width. In this paper, we consider the problem of preprocessing a graph of size
n
and clique-width
k
to build space-efficient data structures that support basic graph navigation queries. First, by way of a counting argument, which is of interest in its own right, we prove the space requirement of any representation is
Ω
(
k
n
)
. Then we present a navigation oracle which answers adjacency query in constant time and neighborhood queries in constant time per neighbor. This oracle uses
O
(
kn
) space (i.e.,
O
(
kn
) bits). We also present a degree query which reports the degree of each given vertex in
O
(
k
log
∗
(
n
)
)
time using
O
(
k
n
log
∗
(
n
)
)
bits.