 Since a graph consists of a set of vertices V and edges E, we can also consider subsets of vertices and edges. A subgraph is a subset of vertices V prime and edges E prime, where V prime includes all vertices incident on an edge in E prime. So, if our graph consists of the vertices and edges, we can take some subset of vertices and edges, and again the key requirement is that if we have an edge in our subgraph, we have to include the incident vertices. Although note that there's no requirement, we have to include edges incident on any particular vertex. Now in principle, we can select any edges we want along with all their incident vertices, but in practice two subgraphs are of importance. The induced graph is a subset of vertices V prime with all edges incident on these vertices, and the edge induced graph is a subset of edges E prime with all vertices incident on these edges. Note that while we could use the term vertex induced subgraph for the first one, we don't. As a general rule, unless otherwise specified, base things on the vertices. So we might try to find an induced graph and an edge induced graph that is not an induced graph. So for the induced graph, we'll select any vertices we want, how about these, and include all the incident edges. For a subgraph not to be induced, it can't include all the edges of its chosen vertices. So let's deselect a few edges and drop the vertices, and that gives us our edge induced subgraph. Given the graph G, a k-clic is a complete subgraph with k vertices. For obvious reasons, a three-clic is often called a triangle. For example, we might try to find any three-clics in the graph shown. Do four-clics exist? How about five-clics? So a three-clic would be three points each connected with the others. We can find a few of them. A four-clic would require four points each of which is connected to the other three. Since every vertex in a four-clic would have to be connected to at least three other vertices, we can ignore any vertices that are only connected to one or two others, as well as any edges incident on those vertices. We can then try to find a four-clic among the remaining subgraph. Likewise, a five-clic could only use vertices of degree four. However, we would need five of them. And if we look carefully at the graph, we see that there's only three vertices with degree four or more. And so there is no five-clic. And this leads to the following. Given a graph G, we can ask several questions about it. Does it have a k-clic? How many? What's the largest k-clic in the graph? Fortunately, these problems are extremely hard, which means that you could be the first to find a solution.