A note on clique-width and tree-width for structures

TL;DR

This paper demonstrates that a naive extension of clique-width to arbitrary structures can be unbounded even when the structures have bounded tree-width, and proposes a correction involving element fusion.

Contribution

It provides a simple proof of the unboundedness issue and introduces a correction method using element fusion for clique-width in structures.

Findings

Naive clique-width extension can be unbounded on bounded tree-width structures

Fusion of elements corrects the unboundedness issue

Simplifies understanding of clique-width in complex structures

Abstract

We give a simple proof that the straightforward generalisation of clique-width to arbitrary structures can be unbounded on structures of bounded tree-width. This can be corrected by allowing fusion of elements.