Closure of an Undirected Graph

In the first graph, no more edges can be added, because the pairs $(v_1,v_2)$ and $(v_1,v_4)$ have degree sum $3$, and every other pair is already adjacent. So this graph is its own closure.

For the second graph, we can add edges $v_1v_3$ and $v_2v_4$ immediately, since for both pairs the degree sum is $4$. Now the degree sum of $v_1,v_4$ is $5$, so we can add this edge too (note that we could not add this edge initially). There are no more non-adjacent pairs, so we've reached the closure.

The point of this definition is that a simple graph is Hamiltonian if and only if its closure is Hamiltonian - this is the Bondy-Chvátal theorem.