Metric dimension on two-connected series–parallel graphs

Determine whether Metric Dimension is polynomial-time solvable on two-connected graphs of treewidth two, equivalently on two-terminal series–parallel graphs.

Background

The paper’s NP-completeness construction uses graphs of treewidth two, but the constructed graph becomes two-terminal series–parallel after removing the triangle attached at the terminal t_m. The triangle is essential to the lower-bound argument because its twin vertices force a landmark adjacent to t_m. This leaves unresolved whether the apparent source of hardness persists in two-connected series–parallel graphs themselves.