Query efficient implementation of graphs of bounded clique-width

Abstract

If P( x 1,…, x k ) is a graph property expressible in monadic second-order logic, where x 1,…, 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 x 1,…, x k of G, one can decide whether P( x 1,…, x k ) holds in time O(log( n)) by using only I( x 1),…, I( x k ). The preprocessing can be done in time O ( n log ( n ) ) . One can do the same for any fixed monadic second-order optimization function (like distance) by using information of size O(log 2( n)) for each vertex and computation time O(log 2( n)). In this case preprocessing time is O(–log 2( 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.