28.1.2. Definition of a Vertex Cut

So, let us start with the definition of a vertex cut. It is also called as a separating set. So, imagine you are given a graph G = (V, E). Then a proper subset V’ ⊆ V of the set of vertices is called the vertex cut if removing the vertices in V’ disconnect your graph.

A vertex cut is a specific group of vertices in a graph whose removal causes the graph to become disconnected. To understand this, visualize a network represented as a graph, where the dots are connected by lines. If we remove certain dots (vertices), we might break the connections between the remaining dots, leading to isolated groups. This set of dots is what we call a vertex cut.

Consider a road map of a city where intersections (vertices) are connected by roads (edges). If a particular intersection is closed (removal of a vertex), could that temporarily separate parts of the city (disconnected graph)? The intersections that you close represent the vertex cut.

So, remember if your graph has an articulation point then that articulation point itself can constitute a V’ of potential V’ whose deletion will disconnect the graph.

An articulation point is a vertex that, if removed, increases the number of connected components of the graph. This means it plays a crucial role in keeping parts of the graph connected. If such a point is present, it can single-handedly serve as a vertex cut by itself, thereby proving to be critical in the structure of the graph.

Think of a bridge connecting two islands. If that bridge (articulation point) is destroyed, the islands become isolated from each other. Thus, the bridge is a vital connection point and represents a vertex cut.

But it might be possible that your graph may not have an articulation point in which case you may need to delete more than one vertex in the graph to disconnect it.

In some graphs, there may be no single vertex whose removal will disconnect the graph. Instead, we might need to remove multiple vertices. This implies a more complex connectivity structure within the graph than in cases where an articulation point exists.

Imagine a tightly knit group of friends where each friend represents a vertex. If some friendships (connections) are broken, but no single friend holds a key role, you might need to remove several friendships to separate everyone into different groups. This reflects the need to remove multiple vertices to cause disconnection.

So, if I take this graph and if I remove the nodes c and node f, then this node g will get disconnected from the network, because these edges will also go away.

Let's visualize a graph where nodes represent cities and edges represent highways. By removing certain cities (nodes c and f), we cut off the highways leading to city g, making it isolated. This explicitly showcases how certain combinations of vertex removals can lead to disconnections within the graph.

Consider a network of water pipes distributing water to various households in a neighborhood. If two essential junctions (c and f) are closed, a third household (g) may find itself without water supply. The removal of junctions is akin to a vertex cut.

So, now the question is, can I say that every connected graph which has nodes as a cut vertex and the answer is yes, except when the graph is a complete graph.

In most cases, connected graphs have vertices such that their removal will lead to disconnections; however, this doesn't apply to complete graphs. In complete graphs, every vertex connects to every other vertex. Thus, the removal of any vertex still leaves the remaining vertices fully connected, so you can't form a vertex cut unless you remove almost all vertices.

Imagine a complete party where everyone knows each other. If one person leaves, the remaining guests still can all talk and interact. It’s only when most attendees leave that some groupings will be formed.

So, here as of now there is no criteria on the cardinality of V’, |V’|. I am just checking with a V’ constitutes a vertex cut or not whether deleting the vertices in V’ disconnects the graph or not.

The size of the vertex cut, denoted as |V’|, doesn't have strict limits in the basic definition. You may have a small set of vertices that when removed, disconnect the graph. Therefore, analyzing whether a collection of vertices forms a vertex cut is more important than how many vertices are in that collection.

Consider a network of computers in an office. Removing a single computer might not disrupt anything, but cutting the connections to certain key machines (the vertex cut) can cause the entire network to become disconnected.

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