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A graph is called k-vertex-connected or k-connected if its vertex connectivity is k or greater. More precisely, any graph G (complete or not) is said to be k -vertex-connected if it contains at least k + 1 vertices, but does not contain a set of k − 1 vertices whose removal disconnects the graph; and κ ( G ) is defined as the largest k such ...
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.
A directed graph is weakly connected (or just connected [9]) if the undirected underlying graph obtained by replacing all directed edges of the graph with undirected edges is a connected graph. A directed graph is strongly connected or strong if it contains a directed path from x to y (and from y to x) for every pair of vertices (x, y). The ...
However, if squares are only partially braced (for instance by connecting two opposite corners by a string or wire that prevents expansive motion but does not prevent contractive motion), then the graph is directed, and represents a rigid structure if and only if it is a strongly connected graph. [6] An associated strong connectivity ...
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.
Similarly, a strict partial order that is connected is a strict total order. A relation is a total order if and only if it is both a partial order and strongly connected. A relation is a strict total order if, and only if, it is a strict partial order and just connected. A strict total order can never be strongly connected (except on an empty ...
In graph theory, Robbins' theorem, named after Herbert Robbins (), states that the graphs that have strong orientations are exactly the 2-edge-connected graphs.That is, it is possible to choose a direction for each edge of an undirected graph G, turning it into a directed graph that has a path from every vertex to every other vertex, if and only if G is connected and has no bridge.
5. A strongly perfect graph is a graph in which every induced subgraph has an independent set meeting all maximal cliques. The Meyniel graphs are also called "very strongly perfect graphs" because in them, every vertex belongs to such an independent set. subforest A subgraph of a forest. subgraph