Definable decompositions for graphs of bounded linear cliquewidth

TL;DR

This paper establishes a method to produce bounded-width decompositions for graphs with bounded linear cliquewidth using MSO_1-transductions, linking definability and recognizability.

Contribution

It introduces an MSO_1-transduction that nondeterministically outputs decompositions for graphs of bounded linear cliquewidth, proving their definability and recognizability equivalence.

Findings

Existence of MSO_1-transduction for bounded linear cliquewidth graphs

Equivalence of CMSO_1-definability and recognizability in this class

Decomposition width bounded by a function of k

Abstract

We prove that for every positive integer k, there exists an MSO_1-transduction that given a graph of linear cliquewidth at most k outputs, nondeterministically, some cliquewidth decomposition of the graph of width bounded by a function of k. A direct corollary of this result is the equivalence of the notions of CMSO_1-definability and recognizability on graphs of bounded linear cliquewidth.