Computing or bounding Hankel and circuit rank

Determine how to compute, or derive general upper bounds for, the Hankel rank or circuit rank associated with an arbitrary graph property P, and characterize the information about P sufficient to perform such estimates or computations.

Background

Finite Hankel rank or circuit rank is the paper’s purely combinatorial substitute for logical definability in generalized Courcelle-type theorems. When the relevant rank is finite, the main theorem provides a finite look-up table and a fixed-parameter tractable recognition procedure for structures presented by suitable parse trees.

The paper explicitly states that the rank is not generally computable or even known to admit a general upper bound. It identifies the investigation of what information about a graph property permits rank estimation or computation as a challenging project, while noting that simple examples are supplied earlier through the computations for standard graph operations and the continuum family of low-rank properties.