Identify further parameters coarsely equivalent to path-length

Identify additional graph parameters, beyond those studied in the paper, that are coarsely equivalent to the path-length of a graph.

Background

The paper presents a broad collection of parameters that are bounded within constant factors of path-length, including decomposition, embedding, graph-power, domination, and minor-fatness parameters. It explicitly asks whether this family can be extended by other meaningful graph parameters.

Any proposed parameter would need to be shown to be bounded above and below by universal constant multiples of path-length over all graphs.