Abstract

If P(x₁,…,x_k) is a graph property expressible in monadic second-order logic, where x₁,…,x_k denote vertices, if G is a graph with n vertices and of clique-width at most p where p is fixed, then we can associate with each vertex u of G a piece of information I(u) of size O(log(n)) such that, for all vertices x₁,…,x_k of G, one can decide whether P(x₁,…,x_k) holds in time O(log(n)) by using only I(x₁),…,I(x_k). The preprocessing can be done in time O(nlog(n)).

One can do the same for any fixed monadic second-order optimization function (like distance) by using information of size O(log²(n)) for each vertex and computation time O(log²(n)). In this case preprocessing time is O(–log²(n)).

Clique-width is a complexity measure on graphs similar to tree-width, but more powerful since every set of graphs of bounded tree-width has bounded clique-width, but not conversely.

Similar results apply to graphs of tree-width at most w and to properties and functions expressed in the version of monadic second-order logic allowing quantifications on sets of edges.

Article Outline

1. Introduction

2. Definitions

2.1. Monadic second-order logic

2.2. Graphs

2.3. MS transduction

2.4. An example of MS transduction

3. Monadic second-order queries on trees

4. Balanced trees and terms

5. Efficient implementation of graph queries

5.1. Main theorems

5.2. Monadic second-order logic with edge set quantifications

5.3. The special case of distance

6. Conclusion

References