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Type of graph
biconnected graph on four vertices and four edges A graph that is not biconnected. The removal of vertex x would disconnect the graph. A biconnected graph
Biconnected_graph
Maximal biconnected subgraph
In graph theory, a biconnected component or block (sometimes known as a 2-connected component) is a maximal biconnected subgraph. Any connected graph decomposes
Biconnected_component
Graph whose biconnected components are all cliques
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
Block_graph
Non-crossing graph with vertices on outer face
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
Outerplanar_graph
Index of articles associated with the same name
(graph theory), a cycle in a graph Forest (graph theory), an undirected graph with no cycles Biconnected graph, an undirected graph in which every edge belongs
Cyclic_graph
Path in a graph that visits each vertex exactly once
Hamiltonian graphs are biconnected, but a biconnected graph need not be Hamiltonian (see, for example, the Petersen graph). An Eulerian graph G (a connected
Hamiltonian_path
Geometric graph with unit edge lengths
graphs (or subgraphs thereof) at a vertex produce strict (respectively non-strict) unit distance graphs, every forbidden graph is a biconnected graph
Unit_distance_graph
Natural number
factors. 294 is the number of planar biconnected graphs with 7 vertices. Biconnected graphs are two dimensional graphs with a given number of points and
294_(number)
size. biclique Synonym for complete bipartite graph or complete bipartite subgraph; see complete. biconnected Usually a synonym for 2-vertex-connected, but
Glossary_of_graph_theory
Graph representing faces of another graph
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
Dual_graph
Partition of a graph whose components are reachable from all vertices
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
Strongly_connected_component
On graph coloring and neighborhood size
theorem. If the graph is not biconnected, its biconnected components may be colored separately and then the colorings combined. If the graph has a vertex
Brooks'_theorem
Graph representing edges of another graph
odd length greater than three. Equivalently, a graph is line perfect if and only if each of its biconnected components is either bipartite or of the form
Line_graph
width one. The biconnected decomposition of an arbitrary constraint satisfaction problem is the biconnected decomposition of its primal graph. Every constraint
Decomposition method (constraint satisfaction)
Decomposition_method_(constraint_satisfaction)
Graph whose line graph is perfect
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
Line_perfect_graph
Graph with tight clique-coloring relation
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
Perfect_graph
Natural number
178 biconnected graphs with six vertices, among which one is designated as the root and the rest are unlabeled. There are also 178 median graphs on nine
178_(number)
Graph with a prism as its skeleton
Hamiltonian cycle. even sided prism graphs are bipartite graphs. Among all biconnected cubic graphs, the prism graphs have within a constant factor of the
Prism_graph
Representation of a graph's triconnected components
In graph theory, a branch of mathematics, the triconnected components of a biconnected graph are a system of smaller graphs that describe all of the 2-vertex
SPQR_tree
Mathematical graph relating to chess
8} board. There is a Hamiltonian cycle for each queen's graph, and the graphs are biconnected (they remain connected if any single vertex is removed)
Queen's_graph
Edge whose deletion would disconnect a graph
intersection (the Eulerian graphs are both bridgeless and almost-Eulerian), but they do not contain each other. Biconnected component Cut (graph theory) Bollobás
Bridge_(graph_theory)
Gluing graphs at complete subgraphs
as a k-clique-sum of smaller graphs. For instance, the SPQR tree of a biconnected graph is a representation of the graph as a 2-clique-sum of its triconnected
Clique-sum
Adjacent subset of an undirected graph
cluster graph is a graph whose connected components are cliques. A block graph is a graph whose biconnected components are cliques. A chordal graph is a
Clique_(graph_theory)
Graph whose shortest paths are unique
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
Geodetic_graph
Graph representing intersections between given sets
intersection graph of maximal cliques of another graph A block graph or clique tree is the intersection graph of biconnected components of another graph Scheinerman
Intersection_graph
Graph which remains connected when k or fewer nodes removed
complete graph Kn. A k-connected graph is by definition connected; it is called biconnected for k ≥ 2 and triconnected for k ≥ 3. Every graph decomposes
Vertex_connectivity
Maximal subgraph whose vertices can reach each other
other forms of graph connectivity, including the weak components and strongly connected components of directed graphs and the biconnected components of
Component_(graph_theory)
Trail in which only the first and last vertices are equal
Bipartite graph, a graph without odd cycles (cycles with an odd number of vertices) Cactus graph, a graph in which every nontrivial biconnected component
Cycle_(graph_theory)
Graph orientation with one source and sink
the other direction, if the graph is not 2-vertex-connected, then it has an articulation vertex v separating some biconnected component of G from s and
Bipolar_orientation
Characteristic of undirected graphs
graph theory, a branch of mathematics, the rank of an undirected graph has two unrelated definitions. Let n equal the number of vertices of the graph
Rank_(graph_theory)
Edges that hit all cycles in a graph
connected component of the given graph, and to break these strongly connected components down even farther to their biconnected components by splitting them
Feedback_arc_set
Recursively-formed graph with two terminal vertices
most 2, if and only if every biconnected component is a series–parallel graph. The maximal series–parallel graphs, graphs to which no additional edges
Series–parallel_graph
Intersection graph of unit intervals on the real line
Hamiltonian path. An indifference graph has a Hamiltonian cycle if and only if it is biconnected. An indifference graph obeys the reconstruction conjecture:
Indifference_graph
Mathematical problem of making a structure rigid
the rigid solutions correspond to biconnected graphs; for tension bracing, they correspond to strongly connected graphs. In both cases, the minimal solutions
Grid_bracing
of W. T. Tutte on the existence of nowhere-zero 5-flows in biconnected undirected graphs is true, this bound would improve to ⌊ k / 5 ⌋ {\displaystyle
Woodall's_conjecture
Graph drawing with vertices on a circle
of vertices within a larger graph drawing, such as its biconnected components, clusters of genes in a gene interaction graph, or natural subgroups within
Circular_layout
Fewest graph edges whose removal breaks all cycles
In any biconnected graph with circuit rank r {\displaystyle r} , every open ear decomposition has exactly r {\displaystyle r} ears. A graph with cyclomatic
Cyclomatic_number
t in a directed graph, if t is reachable from s. Formally, the decision problem is given by PATH = {⟨D, s, t⟩ | D is a directed graph with a path from
St-connectivity
Set of graph nodes which separate a given pair of nodes if removed
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
Vertex_separator
Mathematical graph theorem
Diks, Krzysztof; Stanczyk, Piotr (2010), "Perfect matching for biconnected cubic graphs in O(n log2 n) time", in van Leeuwen, Jan; Muscholl, Anca; Peleg
Petersen's_theorem
Graph layout on multiple half-planes
"Testing the simultaneous embeddability of two graphs whose intersection is a biconnected or a connected graph", Journal of Discrete Algorithms, 14: 150–172
Book_embedding
Graph cycle which does not separate remaining elements
rank less than three (such as a cycle graph or theta graph) every cycle is peripheral, but every biconnected graph with circuit rank three or more has a
Peripheral_cycle
Topics referred to by the same term
representation theory Block, in graph theory, is a biconnected component, a maximal biconnected subgraph of a graph Aschbacher block of a finite group
Block
series–parallel graphs are a subfamily of the partial 2-trees, and more strongly a graph is a partial 2-tree if and only if each of its biconnected components
Partial_k-tree
1990 book by Gerhard Ringel and Nora Hartsfield
algebraic graph theory and spectral graph theory, connectivity of a graph (or even biconnected components), Hall's marriage theorem, line graphs, interval
Pearls_in_Graph_Theory
configurations to handle decompositions of biconnected components. For almost trees of bounded degree (graphs where each biconnected component has at most a constant
Maximum_common_edge_subgraph
for structural analysis of biconnected flow graphs. The triconnected components of the undirected version of a flow graph are shown to be useful for discovering
Program_structure_tree
Assignment of colors to edges of a graph
search over all possible assignments of colors to edges. Every biconnected 3-regular graph with n vertices has O(2n/2) 3-edge-colorings; all of which can
Edge_coloring
Algorithmic problem of finding non-crossing drawings
In graph theory, the planarity testing problem is the algorithmic problem of testing whether a given graph is a planar graph (that is, whether it can
Planarity_testing
Vishkin (2012a, 2012b) for the Graph Connectivity (Connectivity (graph theory)), Graph Biconnectivity (biconnected graph) and Graph Triconnectivity (Triconnected
Explicit_multi-threading
pairs of outerplanar graphs, and for Biconnected graphs, i.e. pairs of graphs whose intersection is biconnected. Any two planar graphs can have a simultaneous
Simultaneous_embedding
Graph formed by subdivision of triangles
been used for the biconnected components of a graph that is not itself biconnected. An Apollonian network is a maximal planar graph in which all of the
Apollonian_network
Sliding puzzle with fifteen pieces and one space
puzzle on each of the biconnected components of that vertex. Excluding these cases, Wilson showed that other than one exceptional graph on 7 vertices, it
15_puzzle
Size of largest complete graph made by contracting edges of a given graph
in G. A graph has Hadwiger number at most three if and only if its treewidth is at most two, which is true if and only if each of its biconnected components
Hadwiger_number
Connectivity measure in graph theory
In graph theory, the cycle rank of a directed graph is a digraph connectivity measure proposed first by Eggan and Büchi (Eggan 1963). Intuitively, this
Cycle_rank
Hierarchical clustering of graph edges
rather than simple graphs are considered) and the three-edge path graph. The graphs of branchwidth 2 are the graphs in which each biconnected component is a
Branch-decomposition
Cycle rank Rank (graph theory) SPQR tree St-connectivity Pixel connectivity Vertex separator Strongly connected component Biconnected graph Bridge v t e
Pixel_connectivity
Mathematical method in graph theory
aggregate information on subtrees. Tarjan, R.E.; Vishkin, U. (1984). Finding biconnected components and computing tree functions in logarithmic parallel time
Euler_tour_technique
Any planar graph can be subdivided by removing a few vertices
smaller size at the expense of a more uneven partition of the graph. In biconnected planar graphs that are not maximal, there exist simple cycle separators
Planar_separator_theorem
Topics referred to by the same term
practices for their own use Articulation point, in graph theory, shared vertices of a biconnected component Articulatory suppression, a process of inhibiting
Articulation
Representation of a graph as a path graph "thickened" by some amount
dual graph must be within a constant factor of each other: bounds of this form are known for biconnected outerplanar graphs and for polyhedral graphs. For
Pathwidth
Graph algorithm
In graph theory, the strongly connected components of a directed graph may be found using an algorithm that uses depth-first search in combination with
Path-based strong component algorithm
Path-based_strong_component_algorithm
Design technique for parallel algorithms
forest of rooted trees, connected components, minimum spanning trees, and biconnected components. However, pointer jumping has also shown to be useful in a
Pointer_jumping
American computer scientist
50, 2, 1995, 259-273. "Path-based depth-first search for strong and biconnected components," H.N. Gabow, Information Processing Letters 74, 2000, 107-114
Harold_N._Gabow
Israeli-American computer scientist
ancestor, connected components, spanning trees, biconnected components, Euler tours in trees and graphs, strong orientation, triconnected components, ear
Uzi_Vishkin
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