Graph Bridge

A bridge of a connected graph is a graph edge whose removal disconnects the graph (Chartrand 1985, p. 45; Skiena 1990, p. 177). More generally, a bridge is an edge of a not-necessarily-connected graph  whose removal increases the number of components of  (Harary 1994, p. 26; West 2000, p. 23).

An edge of a connected graph is a bridge iff it does not lie on any cycle. A bridge therefore cannot be a cycle chord.

A bridge is also known as an isthmus, cut-edge (West 2000, p. 23), or cut arc.

Every edge of a tree is a bridge. A connected cubic graph contains a bridge iff it contains an articulation vertex (Skiena 1990, p. 177), i.e., if it is not a biconnected graph.

A graph containing one or more bridges is said to be a bridged graph, while a graph containing no bridges is called a bridgeless graph.

The Wolfram Language function FindEdgeCut[g] returns an edge cut of smallest size for a graph, which corresponds to a graph bridge if the set is of size 1. Precomputed bridges for many named graphs can be listed using GraphData[graph, "Bridges"].

The analog of a graph bridge for vertices is called an articulation vertex.

REFERENCES

Chartrand, G. "Cut-Vertices and Bridges." §2.4 in Introductory Graph Theory. New York: Dover, pp. 45-49, 1985.

Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994

Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 171 and 177, 1990.

West, D. B. Introduction to Graph Theory, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, pp. 155-158, 2000.