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How is this Lewis dot diagram for CLO3 determined?
Got it. And then what?
Next, you would connect each atom to the central atom with a single bond, and then distribute the remaining electrons as lone pairs around each atom until each atom has a full octet (or duet, in the case of hydrogen). For CLO3, the Lewis dot diagram would look like this:

O
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C-O-Cl   <-- single bonds
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O