Graph Theory Questions Long
A connected graph is a type of graph in which there is a path between every pair of vertices. In other words, it is a graph where there are no isolated vertices or disconnected components.
Formally, a graph G is said to be connected if for every pair of vertices u and v in G, there exists a path from u to v. This means that starting from any vertex, we can reach any other vertex in the graph by following a sequence of edges.
To determine if a graph is connected, we can use various algorithms such as depth-first search (DFS) or breadth-first search (BFS). These algorithms traverse the graph and check if all vertices are visited, indicating that the graph is connected. If there are any unvisited vertices after the traversal, then the graph is not connected.
Connected graphs have several important properties and applications in graph theory. For example, they are used in network analysis, where the connectivity of nodes represents the ability to transmit information or resources. Connected graphs also play a crucial role in the study of graph algorithms, as many algorithms are designed specifically for connected graphs.
In summary, a connected graph is a graph where there is a path between every pair of vertices, ensuring that all vertices are reachable from any starting vertex.