Covering lattice with circles

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There is a $m \times n$ lattice and some circles are covering the vertices on it. (The one that's covering the edges is it's boundary not the inside of the circle) How many circles do we need to cover all the edges in the lattice? (I'm sorry I'm not so good at English but I hope you guys understood what I meant to say) I want to ask if there are some ideas or researches done on this subject or some keywords for searching.