1.8. Graph Connectivity

Once we have the definition of a path, let us now define what we call connected graphs and components, so an undirected graph is called connected if there exists a path between every pair of distinct vertices, I stress here between distinct vertices, I am not interested to check whether there exists a path from a node to itself, but between every pair of distinct vertices there should exist at least one path, there could be multiple paths as well but at least one path should be there between every pair of distinct vertices.

A path in graph theory is a sequence of edges that connects two vertices. In the context of connected graphs, this means that for the graph to be considered connected, there must be at least one path between every pair of distinct vertices. Importantly, we are not concerned about whether a vertex can connect back to itself; our focus is solely on connections between different vertices. If such paths exist for all pairs without exception, then the graph is deemed connected.

Imagine a city's road network where each intersection is a vertex and each road between intersections is an edge. If you can drive from one intersection to any other intersection without being blocked, then the city's road network is connected. If there are intersections that you cannot reach from others due to dead ends or blockages, then that part of the city is disconnected.

Now what is the connected component of a graph? A connected component of a graph is the maximal connected subgraph of the graph. What does the maximal connected subgraph mean? A connected subgraph of the graph G and it is maximal in the sense that it cannot be further extended. What does that mean? It means you cannot have another subgraph of the graph G which is also connected such that the connected subgraph is a proper subgraph of that connected subgraph.

Connected components refer to subsets of the graph where each subset is itself connected. A connected component is characterized as being maximal, which means that it includes all the vertices that are reachable from any vertex within that component. If you try to add any other vertex to this subset from outside of it, you would break the property of connectedness. In practical terms, if you have several separate clusters of points in a graph and none of those points connect to any points in another cluster, each cluster represents its own connected component.

Think of connected components like neighborhoods in a city. Each neighborhood is interconnected through local streets (the edges), and as long as you are within that neighborhood, you can reach any home (vertex) from another. However, you cannot reach a home in a different neighborhood without exiting to the main roads (or edges) that connect them, thus those neighborhoods are distinct connected components.

Now let us next define cut vertex and cut edge. So, cut vertices are also called as articulation point or critical vertices. If I take this graph G, the node c is very critical here, because if I delete this node c then it will disconnect the entire graph.

A cut vertex is a vertex in a graph that, when removed, increases the number of connected components. Essentially, its removal splits the graph into two or more separate groups of vertices that are no longer connected to each other. This concept highlights the importance of certain vertices in maintaining the connectivity of the graph.

Consider a bridge in a network of islands. If that bridge (the cut vertex) is destroyed, the islands on either side would no longer be connected, just like removing a cut vertex disconnects parts of a graph. Each island represents a cluster of vertices, and the bridge is critical for connectivity.

Similarly, I can define a critical edge which is also called as a bridge or cut edge. If I take the same graph as above and focus on this edge c and that connecting the node c and f, then the edge connecting the node c and f is very critical because if I delete that edge then the whole graph gets disconnected.

A cut edge, or bridge, is an edge in a graph that, when removed, disconnects the graph into two or more pieces. This indicates that the edge serves as a critical link between two parts of the graph and its absence would mean that those parts can no longer reach each other.

Think about a single road that connects two towns. If that road (the cut edge) is closed due to construction or a landslide, the towns can no longer access one another, leading to isolation until the road is repaired. This emphasizes the role of that specific edge in ensuring connectivity.

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