Math 2.0

Combinatorics Theorem

Characterization of bridges

Builds on CycleBridge

Statement

Let GG be a graph and let ee be an edge of GG. Then ee is a bridge of GG if and only if ee is not an edge of any cycle of GG.

Remarks

Proof idea: let e=uve = uv. If ee lies on a cycle, the rest of the cycle is a path from uu to vv in G−eG - e. Conversely, a path from uu to vv in G−eG - e together with ee gives a cycle. A consequence: a graph is a forest if and only if every edge is a bridge.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

Show the path as a graph
Graph, definition, no evidence levelGraphdefinitionWalk, definition, no evidence levelWalkdefinitionReachability, definition, no evidence levelReachabilitydefinitionTrail, definition, no evidence levelTraildefinitionConnected component, definition, no evidence levelConnected componentdefinitionCircuit, definition, no evidence levelCircuitdefinitionBridge, definition, no evidence levelBridgedefinitionCycle, definition, no evidence levelCycledefinitionCharacterization of bridges, statement, no evidence levelCharacterization of b…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Graph definition
  2. Walk definition
  3. Reachability definition
  4. Trail definition
  5. Connected component definition
  6. Circuit definition
  7. Bridge definition
  8. Cycle definition
  9. Characterization of bridges statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Bridge_(graph_theory), lead section
About this page
Tag0FHM8DB6PM
URIm2://0FHM8DB6PM
Kindstatement, theorem
WritingW0 stub
MSC05C40
StatusThis copy of the site has no build data. Run m2 status 0FHM8DB6PM.
UsesCycle, Bridge

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