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DRC - Design Rule Checks - Ideally this is a manufacturability issue. Violations such as spacing etc comes in this.
For example, if there are two nets that are too close (Lesser than the min spacing defined in library), the design may be at present clean with no shorts. But there is a possibility of these two being shorted when manufactured.
LVS - Layout Vs Schematic. - Shorts and opens in general.
Does your chip function as the input verilog netlist.
If in netlist, there are two nets A and B while A goes as an input to AND and B goes to OR. Suppose because of congestion there exists a short between A and B, then B goes to AND and OR and A goes to AND and OR gates. Hence the circuit will not perform as expected and hence this is an LVS Issue
I have given in a broader way. DRC and LVS are more complicated than spacing and short. But these serve as a better starting point.
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