For every connected graph, the distance-edge-monitoring number is at most the cyclomatic number plus one, and similar linear bounds hold for metric dimension, geodetic number, and isometric path covers.
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Distance-based (and path-based) covering problems for graphs of given cyclomatic number
For every connected graph, the distance-edge-monitoring number is at most the cyclomatic number plus one, and similar linear bounds hold for metric dimension, geodetic number, and isometric path covers.