Trees and Forests
A graph with nodes can contain at most bridges, since adding additional edges must create a cycle. The graphs with exactly bridges are exactly the trees, and the graphs in which every edge is a bridge are exactly the forests.
In every undirected graph, there is an equivalence relation on the vertices according to which two vertices are related to each other whenever there are two edge-disjoint paths connecting them. (Every vertex is related to itself via two length-zero paths, which are identical but nevertheless edge-disjoint.) The equivalence classes of this relation are called 2-edge-connected components, and the bridges of the graph are exactly the edges whose endpoints belong to different components. The bridge-block tree of the graph has a vertex for every nontrivial component and an edge for every bridge.
Read more about this topic: Bridge (graph Theory)
Famous quotes containing the words trees and, trees and/or forests:
“Any walk through a park that runs between a double line of mangy trees and passes brazenly by the ladies’ toilet is invariably known as “Lover’s Lane.””
—F. Scott Fitzgerald (1896–1940)
“Usually the scenery about them is drear and savage enough; and the logger’s camp is as completely in the woods as a fungus at the foot of a pine in a swamp; no outlook but to the sky overhead; no more clearing than is made by cutting down the trees of which it is built, and those which are necessary for fuel.”
—Henry David Thoreau (1817–1862)
“‘Tis chastity, my brother, chastity.
She that has that is clad in complete steel,
And like a quivered nymph with arrows keen
May trace huge forests and unharbored heaths,
Infamous hills and sandy perilous wilds,
Where, through the sacred rays of chastity,
No savage fierce, bandit, or mountaineer
Will dare to soil her virgin purity.”
—John Milton (1608–1674)