A Verifier for a language $A$ is an algorithm $V$ such that $$A=\left\{ w \space | V \space accepts \space <w,c> \space for \space some \space string \space c\right\}$$$$A=\left\{ w \space | V \space \text{accepts} \space \langle w,c\rangle \space\text{for}\space \text{some} \space \text{string} \space c\right\}.$$
My question about this definition is what is this string $c$ stands for?
Let's say I have a graph and I want to verify if it has a clique. What would be my $c$?
