Infinite graphs with more than one end have infinite strong metric dimension; for graphs with finitely many cycles, weak metric dimension is finite exactly when there are finitely many degree-3 vertices, and strong metric dimension is finite exactly when there is also one end.
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Infinite graphs with finite metric dimension
Infinite graphs with more than one end have infinite strong metric dimension; for graphs with finitely many cycles, weak metric dimension is finite exactly when there are finitely many degree-3 vertices, and strong metric dimension is finite exactly when there is also one end.