In graph theory, a biconnected graph is a connected and "nonseparable" graph, meaning that if any one vertex were to be removed, the graph will remain connected. Therefore a biconnected graph has no articulation vertices.
The property of being 2-connected is equivalent to biconnectivity, except that the complete graph of two vertices is usually not regarded as 2-connected.
This property is especially useful in maintaining a graph with a two-fold redundancy, to prevent disconnection upon the removal of a single edge (or connection).
The use of biconnected graphs is very important in the field of networking (see Network flow), because of this property of redundancy.
biconnectedgraph on four vertices and four edges A graph that is not biconnected. The removal of vertex x would disconnect the graph. A biconnected graph...
In graph theory, a biconnected component (sometimes known as a 2-connected component) is a maximal biconnected subgraph. Any connected graph decomposes...
Hamiltonian graphs are biconnected, but a biconnectedgraph need not be Hamiltonian (see, for example, the Petersen graph). An Eulerian graph G (a connected...
(graph theory), a cycle in a graph Forest (graph theory), an undirected graph with no cycles Biconnectedgraph, an undirected graph in which every edge belongs...
size. biclique Synonym for complete bipartite graph or complete bipartite subgraph; see complete. biconnected Usually a synonym for 2-vertex-connected, but...
In graph theory, a branch of combinatorial mathematics, a block graph or clique tree is a type of undirected graph in which every biconnected component...
a cycle graph). As they showed, when the base graph is biconnected, a graph constructed in this way is planar if and only if its base graph is outerplanar...
In graph theory, a branch of mathematics, the triconnected components of a biconnectedgraph are a system of smaller graphs that describe all of the 2-vertex...
In the mathematical discipline of graph theory, the line graph of an undirected graph G is another graph L(G) that represents the adjacencies between edges...
perfect line graph L ( G ) {\displaystyle L(G)} is a line perfect graph. These are the graphs whose biconnected components are bipartite graphs, the complete...
In the mathematical theory of directed graphs, a graph is said to be strongly connected if every vertex is reachable from every other vertex. The strongly...
mathematical discipline of graph theory, the dual graph of a planar graph G is a graph that has a vertex for each face of G. The dual graph has an edge for each...
intersection graph of maximal cliques of another graph A block graph of clique tree is the intersection graph of biconnected components of another graph Scheinerman...
complete graph is geodetic, and every geodetic subdivided complete graph can be obtained in this way. If every biconnected component of a graph is geodetic...
the graph into three perfect matchings, or equivalently an edge coloring of the graph with three colors. Every biconnected n-vertex cubic graph has O(2n/2)...
cycle is a triangle. A graph is line perfect if and only if each of its biconnected components is a bipartite graph, the complete graph K4, or a triangular...
representation theory Block, in graph theory, is a biconnected component, a maximal biconnected subgraph of a graph Aschbacher block of a finite group...
In graph theory, a vertex subset S ⊂ V {\displaystyle S\subset V} is a vertex separator (or vertex cut, separating set) for nonadjacent vertices a and...
graphs (or subgraphs thereof) at a vertex produce strict (respectively non-strict) unit distance graphs, every forbidden graph is a biconnectedgraph...
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