Metric Dimension parameterized by treewidth is W[1]-hard, with no f(pw)n^{o(pw)} algorithm under ETH on constant-degree graphs.
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Proves dem(G-e) - dem(G) ≤ 2 (sharp), shows arbitrary changes from vertex deletion, and gives an algorithm to verify monitoring-set persistence after edge removal.
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Perturbation results for distance-edge-monitoring numbers
Proves dem(G-e) - dem(G) ≤ 2 (sharp), shows arbitrary changes from vertex deletion, and gives an algorithm to verify monitoring-set persistence after edge removal.