Connected graph

Definition

A connected graph is a type of graph in which there is a path between every pair of vertices. This means you can travel from any vertex to any other vertex without having to leave the graph, which is essential for understanding many graph-related concepts like paths, cycles, and connectivity. Connected graphs help in analyzing various properties such as degree sequences and are crucial when exploring special types of paths and cycles, as well as determining the robustness of a graph's structure.

5 Must Know Facts For Your Next Test

1. In a connected graph, every vertex is reachable from every other vertex, meaning no isolated parts exist within the graph.
2. A connected graph with no cycles is known as a tree, which has special properties related to spanning trees.
3. If you add an edge to a disconnected graph, it may become connected if the new edge links previously unconnected components.
4. In terms of traversal algorithms, connected graphs ensure that depth-first and breadth-first searches will visit all vertices if started from any vertex.
5. The minimum number of edges needed to connect 'n' vertices in a connected graph is 'n-1', which forms the basis for constructing spanning trees.

Review Questions

• How do you determine if a given graph is connected or not?
  • To determine if a graph is connected, check whether there exists a path between every pair of vertices. This can be done using traversal methods like depth-first search or breadth-first search starting from any vertex. If all vertices are reachable by these methods without running out of unvisited vertices, then the graph is connected; otherwise, it is disconnected.
• What role does connectivity play in determining whether a cycle or path exists within a connected graph?
  • Connectivity is fundamental in finding paths and cycles because, in a connected graph, every vertex can be reached from any other vertex. This implies that if you start at one vertex, you can construct paths that may eventually form cycles. For instance, Eulerian and Hamiltonian paths rely on the connectivity of the graph to ensure that you can visit each vertex or edge exactly once or return to the starting point without getting stuck.
• Evaluate how the presence of cut vertices affects the overall connectivity of a connected graph and its applications.
  • Cut vertices significantly impact the connectivity of a connected graph because their removal can split the graph into two or more disconnected components. This characteristic is crucial for analyzing network reliability and vulnerability; for example, if a network's cut vertex fails, it may disconnect critical parts of the network. Therefore, identifying these vertices helps in designing robust networks that can withstand failures while maintaining connectivity.

Related terms

Disconnected Graph: A graph in which at least one pair of vertices does not have a path connecting them, meaning the graph is made up of multiple components.

Path: A sequence of edges that connects a sequence of vertices, with each edge connecting two consecutive vertices in the sequence.

Cut Vertex: A vertex whose removal increases the number of connected components in a graph, thus affecting its connectivity.