Editing Bridge (graph theory) (section)

{{Short description|Edge in node-link graph whose removal would disconnect the graph}}
[[Image:Graph cut edges.svg|thumb|upright=0.85|A graph with 16 vertices and six bridges (highlighted in red)]]
[[Image:Undirected.svg|thumb|upright=0.6|An undirected connected graph with no bridge edges]]

In [[graph theory]], a '''bridge''', '''isthmus''', '''cut-edge''', or '''cut arc''' is an [[Glossary of graph theory#edge|edge]] of a [[Graph (discrete mathematics)|graph]] whose deletion increases the graph's number of [[Connected component (graph theory)|connected components]].<ref>{{citation
 | last = Bollobás | first = Béla | author-link = Béla Bollobás
 | doi = 10.1007/978-1-4612-0619-4
 | isbn = 0-387-98488-7
 | location = New York
 | mr = 1633290
 | page = 6
 | publisher = Springer-Verlag
 | series = Graduate Texts in Mathematics
 | title = Modern Graph Theory
 | url = https://books.google.com/books?id=SbZKSZ-1qrwC&pg=PA6
 | volume = 184
 | year = 1998}}.</ref> Equivalently, an edge is a bridge if and only if it is not contained in any [[Cycle (graph theory)|cycle]]. For a connected graph, a bridge can uniquely determine a [[Cut (graph theory)|cut]]. A graph is said to be '''bridgeless''' or '''isthmus-free''' if it contains no bridges.

This type of bridge should be distinguished from an unrelated meaning of "bridge" in graph theory, a subgraph separated from the rest of the graph by a specified subset of vertices; see [[Glossary of graph theory#bridge|bridge]] in the [[Glossary of graph theory]].