singleedges

Singleedges is a term used in graph theory to describe edges whose removal increases the number of connected components of a graph. In a connected graph, a singleedge is exactly an edge that is not contained in any cycle; such an edge is also known as a bridge or cut-edge. Removing a singleedge from a connected graph always splits it into two components.

Intuitively, a singleedge represents a critical point of failure in a network: it is the sole link between two portions of the graph, and its loss disconnects those portions. Conversely, edges that lie on a cycle are not singleedges, because alternative routes exist within the cycle that maintain connectivity after removing any one of those edges.

Bridges have several structural implications. If all bridges are contracted, the graph decomposes into its 2-edge-connected components, and the resulting structure is a tree called the bridge-tree or block tree. This perspective helps in understanding network reliability and in designing systems with redundancy.

Efficient detection of singleedges is central in graph algorithms. A standard approach uses depth-first search to compute discovery times and low-link values; an edge (u, v) is a bridge if low[v] > disc[u] when v is a tree child of u. This Tarjan-style algorithm runs in linear time O(V + E). For dynamic graphs, incremental or decremental bridge-finding methods exist to maintain bridges under edge insertions or deletions.

Related concepts include edge connectivity, which measures the minimum number of edges needed to disconnect the graph, and 2-edge-connected graphs, which have no bridges. Identifying singleedges is important in network design, reliability analysis, and infrastructure planning.