Compact Representation of Graphs of Small Clique-Width
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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 \(\varOmega (kn)\). 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 \(O(kn \log ^*(n))\) bits.
KeywordsClique-width Navigation oracles Compact representation
The author would like to thank the anonymous reviewers for their careful reading of the paper and for their valuable comments and suggestions which resulted in substantial improvement of the paper.
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