Linear-time dependence in the merge-width model-checking algorithm

Strengthen the first-order model-checking algorithm for graph classes of bounded merge-width, given a witnessing construction sequence, so that its dependence on the number of vertices is linear rather than cubic.

Background

The main algorithm runs in cubic time in the number of vertices, whereas the corresponding results for bounded expansion and bounded twin-width have linear dependence when their respective decompositions are available. The authors explicitly conjecture that the merge-width algorithm can achieve the same linear dependence.