doublyconnected

Doublyconnected is a term used in graph theory to describe a connected graph that remains connected even after the removal of any single vertex. In other words, if you take out any one node from a doublyconnected graph, the remaining graph will still have a path between any two remaining nodes. This property is also known as vertex-connectivity of at least 2.

A graph that is not doublyconnected is said to be decomposable or separable. If a connected graph has a vertex whose removal disconnects the graph, that vertex is called a cut vertex or articulation point. Therefore, a graph is doublyconnected if and only if it has no cut vertices.

The concept of doublyconnected graphs is important in network design and reliability. A network designed using a doublyconnected topology would be more robust, as the failure of a single node would not disrupt communication between other nodes. This property ensures a higher level of fault tolerance.

Examples of doublyconnected graphs include cycles, complete graphs (for n > 1), and grids. A simple path graph with more than two vertices is not doublyconnected because the removal of an interior vertex will separate the graph. The smallest doublyconnected graph is a triangle (a cycle of length 3).