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A graph with 16 vertices and six bridges (highlighted in red) An undirected connected graph with no bridge edges. In graph theory, a bridge, isthmus, cut-edge, or cut arc is an edge of a graph whose deletion increases the graph's number of connected components. [1] Equivalently, an edge is a bridge if and only if it is not contained in any ...
Tarjan's strongly connected components algorithm is an algorithm in graph theory for finding the strongly connected components (SCCs) of a directed graph. It runs in linear time , matching the time bound for alternative methods including Kosaraju's algorithm and the path-based strong component algorithm .
This gives immediately a linear-time 2-connectivity test and can be extended to list all cut vertices of G in linear time using the following statement: A vertex v in a connected graph G (with minimum degree 2) is a cut vertex if and only if v is incident to a bridge or v is the first vertex of a cycle in C – C 1.
A graph G which is connected but not 2-connected is sometimes called separable. Analogous concepts can be defined for edges. In the simple case in which cutting a single, specific edge would disconnect the graph, that edge is called a bridge. More generally, an edge cut of G is a set of edges whose removal renders the graph disconnected.
The tabular representation lends itself well to computational applications. There are different ways to store graphs in a computer system. The data structure used depends on both the graph structure and the algorithm used for manipulating the graph. Theoretically one can distinguish between list and matrix structures but in concrete ...
Several algorithms based on depth-first search compute strongly connected components in linear time.. Kosaraju's algorithm uses two passes of depth-first search. The first, in the original graph, is used to choose the order in which the outer loop of the second depth-first search tests vertices for having been visited already and recursively explores them if not.
Tarjan's algorithm may refer to one of several algorithms attributed to Robert Tarjan, including: Tarjan's strongly connected components algorithm; Tarjan's off-line lowest common ancestors algorithm; Tarjan's algorithm for finding bridges in an undirected graph [1] Tarjan's algorithm for finding simple circuits in a directed graph [2]
An extension of Robbins' theorem to mixed graphs by Boesch & Tindell (1980) shows that, if G is a graph in which some edges may be directed and others undirected, and G contains a path respecting the edge orientations from every vertex to every other vertex, then any undirected edge of G that is not a bridge may be made directed without changing the connectivity of G.