K-vertex-connected graph

A k-vertex-connected graph is like a big playground with lots of kids playing. If we imagine each kid as a vertex (a point on a graph), and each time they hold hands with another kid as a connection (an edge on a graph), we can create a graph that shows how the playground is connected.

A k-vertex-connected graph is a special kind of playground where any k kids can hold hands and still make sure every kid in the playground (every vertex on the graph) can hold hands with at least one other kid (or vertex). This means that even if some kids leave the playground, there will still be a group of k kids left that can hold hands and connect everyone else together.

Think of it like a big spiderweb: if any k strands get cut, the web still won't fall apart because there are enough other strands connecting everything together. This makes k-vertex-connected graphs important in things like computer networks, where losing one node shouldn't cause the whole system to fail.