Bounded embeddable tree-width of two-source regular-expression graph languages

Determine whether every graph language definable by the regular expressions for graphs constructed with at most two sources (as introduced by Doumane) has bounded embeddable tree-width, where embeddable tree-width is the minimum width of a tree decomposition whose backbone is isomorphic to a spanning tree of the graph.

Background

Recent work by Doumane developed a language of regular expressions characterizing the class of Counting Monadic Second Order Logic (CMSO)-definable graphs that can be built using at most two sources. These regular expressions provide a concise, algebraic description of such graph classes.

The paper introduces embeddable tree-width, a strengthening of standard tree-width that requires the backbone of a tree decomposition to be isomorphic to a spanning tree of the graph. This parameter underlies the completeness results proven for tree-verifiable grammars and is central to the characterization of the CMSO-definable graph languages captured by these grammars.

It remains to be established whether the graph languages described by Doumane’s two-source regular expressions uniformly admit bounded embeddable tree-width. While canonical optimal tree decompositions can be extracted from the structure of those graphs, the known canonical decompositions do not correspond to spanning trees, which is required by embeddable tree-width.