 It's useful to keep in mind the following idea. Notation and terminology aren't standardized until a research area is dead. There's no question about what the derivative of a function is, and so the notation for the derivative is always dy dx, unless it's y' or dot y, but in any case it is standardized. Graph theory, on the other hand, is mostly a development of the past century. Other results are being discovered continuously, so notation and terminology are not standardized. Get used to it. And that's why this guy shows up every now and then. Take it, Ralph. One reason notation and terminology change is that we might discover we need to capture more context. We've been using d sub v to indicate the degree of a vertex, but since we've been adding edges to a graph, the same vertex v might be in different, but related, graphs. So we'll start to use the function notation d of v to indicate the degree of a vertex. One advantage is that function notation allows us to include indices. So d of v is the degree if the identity of the graph is unambiguous, but if we have to specify, we can include the graph as an index, d g of v, d h of v. Now previously we found that if we added an edge to g and found it to be Hamiltonian, if the degree sum was at least n, then g itself was Hamiltonian. Now note that our premises are that g plus u v is Hamiltonian, and the degree of u plus the degree of v is greater than or equal to n. Since we need both, we can choose which one we assume and which one we make our antecedent. And this gives us a different version of the same result. This time we'll assume that our degree sum is greater than or equal to n, and then if g plus the edge is Hamiltonian, then g is Hamiltonian. Now it should be obvious that if our original graph is Hamiltonian, adding an edge leaves it Hamiltonian. If we put this together with our previous result, we get a pair of conditionals where the antecedents and consequence have switched places. And so, putting them together, we can combine the two results in an if and only if theorem. But we really shouldn't. That is to say, if and only if theorems are great for textbooks, and they're good for gaining street cred among gangs of roving mathematicians, but they're not quite as useful when you think about mathematics. We should avoid if and only if constructions. We should either state both conditionals, or we could use the and conversely construction. And that theorem suggests that adding edges between vertices where the degree sum is at least n doesn't change whether or not a graph is Hamiltonian. And this led to the idea of closing a graph by adding edges between non-adjays and vertices whose degree sum was at least n. So, for example, we might try that with a graph shown. There are six vertices. Since a and e are non-adjacent, and the degree sum is at least six, we'll add an edge. Note that now the degree of a and e are both four. Since a is non-adjacent to f, and the degree sum is at least six, we'll join a f. Since c is non-adjacent to e, and the degree sum is six, we'll join ce. And the remaining pairs of non-adjacent vertices have degree sums less than six, so we won't join any of them. And so we could say the graph produced this way is the closure of g. Or can we? If we have more than one pair of non-adjacent vertices whose degree sum is greater than or equal to n, we have a choice of how we add edges. So it's conceivable that different choices lead us to different graphs. So suppose we try to close some graph g. We might add edges e1, e2, and so on in that order to produce some graph g1. Or we might make a different set of choices and add edges l1, l2, and so on, also in that order to produce a graph g2. So if we make the same choices, we'll obviously end up with the same graph, so let's assume at some point we make a different choice. So let ek1, the edge between u and v, the first edge we add to g1 that is different from the edge we add to g2. In some sense, this is where we first make a different choice. So up until this point, we've made all the same choices. So let h be g plus all of these common choices. And that means g1 is h plus some extra choices, and g2 is h plus some extra choices, where this k plus first choice is actually different. Now in g1, we would have added the edge ek1 between u and v. But we'd only add an edge if the degree sum was greater than n. So we know that the degree of u in h plus the degree of v in h has to be at least n. So while we didn't add it as the k plus first edge in g2, we would have added it eventually. Unless it's possible that at some point the degree of u, v, and g2 drops below n, but this can't happen. Can it? No, it can't. So this means that every edge we added to g to produce g1 is an edge we would have added to produce g2, if not at one point, then at some later point. Since we can make the same argument about the edges we added to g to produce g2, this means the edges we added are the same, even if we didn't add them in the same order. And that gives us the graph on n vertices formed by repeatedly joining non-adjacent vertices whose degree sum is at least n is unique. And this means that our process of adding edges will always lead to the same final graph, and consequently we can define the closure of the graph. What's important is that because the closure of the graph is unique, then we might be able to infer properties of the graph from the properties of its closure. And you might notice what I did here simultaneously using the term closure, with the notation we're going to use. And that's because I'm a brilliant author and presenter, and it's also generally good style. And that's because this will help you navigate around the different notation and terminology that's used by different authors and researchers. Where was I? Let's see, brilliant author and presenter, oh yeah, there we were. So suppose the process of producing the closure yields the following sequence. So we add edge 1 to produce g1, we add edge 2 to produce g2, and so on, eventually arriving at the closure of the graph. And suppose our final graph is Hamiltonian. Since we produced it from our previous graph by adding some edge where the degree sum was at least n, then our theorem says that the previous graph is also Hamiltonian. And by the same logic, the graph before that is Hamiltonian, and in fact, all of our graphs are Hamiltonian. And this leads to an important result, if the closure of a graph is Hamiltonian, then the graph itself is Hamiltonian. And why not? It should be obvious that if a graph is Hamiltonian, its closure must also be Hamiltonian. And consequently, we can combine our two conditionals and say that g is Hamiltonian if and only if its closure is Hamiltonian, but we'd rather say that if g is Hamiltonian, its closure is Hamiltonian, and conversely.