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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
graph or clique tree is a type of undirected graph in which every biconnected component (block) is a clique. Block graphs are sometimes erroneously called
Block_graph
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
resulting in width one. The biconnected decomposition of an arbitrary constraint satisfaction problem is the biconnected decomposition of its primal graph
Decomposition method (constraint satisfaction)
Decomposition_method_(constraint_satisfaction)
Topics referred to by the same term
modular representation theory Block, in graph theory, is a biconnected component, a maximal biconnected subgraph of a graph Aschbacher block of a finite group
Block
Maximal subgraph whose vertices can reach each other
connected components of directed graphs and the biconnected components of undirected graphs. The number of components of a given finite graph can be used to count
Component_(graph_theory)
Edge whose deletion would disconnect a graph
bridgeless and almost-Eulerian), but they do not contain each other. Biconnected component Cut (graph theory) Bollobás, Béla (1998), Modern Graph Theory, Graduate
Bridge_(graph_theory)
On graph coloring and neighborhood size
simplified proof of Brooks' theorem. If the graph is not biconnected, its biconnected components may be colored separately and then the colorings combined
Brooks'_theorem
Non-crossing graph with vertices on outer face
outerplanar graph is the same as the number of vertices in its largest biconnected component. For this reason finding Hamiltonian cycles and longest cycles in
Outerplanar_graph
Partition of a graph whose components are reachable from all vertices
every vertex is reachable from every other vertex. The strongly connected components of a directed graph form a partition into subgraphs that are strongly
Strongly_connected_component
Graph whose line graph is perfect
biconnected components is a bipartite graph, the complete graph K4, or a triangular book K1,1,n. Because these three types of biconnected component are
Line_perfect_graph
Trail in which only the first and last vertices are equal
number of vertices) Cactus graph, a graph in which every nontrivial biconnected component is a cycle Cycle graph, a graph that consists of a single cycle
Cycle_(graph_theory)
Topics referred to by the same term
own use Articulation point, in graph theory, shared vertices of a biconnected component Articulatory suppression, a process of inhibiting memory by requiring
Articulation
Graph with tight clique-coloring relation
cliques joined at a single vertex, and the block graphs in which each biconnected component is a clique. The threshold graphs are formed from an empty graph
Perfect_graph
Graph which remains connected when k or fewer nodes removed
graphs decompose into a tree of biconnected components. 2-connected graphs decompose into a tree of triconnected components. The 1-skeleton of any k-dimensional
Vertex_connectivity
Graph algorithm
Harold N. (2000), "Path-based depth-first search for strong and biconnected components" (PDF), Information Processing Letters, 74 (3–4): 107–114, doi:10
Path-based strong component algorithm
Path-based_strong_component_algorithm
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
Feedback_arc_set
Recursively-formed graph with two terminal vertices
and only if it has branchwidth at most 2, if and only if every biconnected component is a series–parallel graph. The maximal series–parallel graphs,
Series–parallel_graph
Graph representing edges of another graph
Equivalently, a graph is line perfect if and only if each of its biconnected components is either bipartite or of the form K4 (the tetrahedron) or K1,1
Line_graph
Mathematical method in graph theory
information on subtrees. Tarjan, R.E.; Vishkin, U. (1984). Finding biconnected components and computing tree functions in logarithmic parallel time. Proceedings
Euler_tour_technique
Graph representing intersections between given sets
graph A block graph or clique tree is the intersection graph of biconnected components of another graph Scheinerman (1985) characterized the intersection
Intersection_graph
Graph whose shortest paths are unique
every biconnected component of a graph is geodetic then the graph itself is geodetic. In particular, every block graph (graphs in which the biconnected components
Geodetic_graph
complete. biconnected Usually a synonym for 2-vertex-connected, but sometimes includes K2 though it is not 2-connected. See connected; for biconnected components
Glossary_of_graph_theory
Sliding puzzle with fifteen pieces and one space
articulation vertex, the problem reduces to the same puzzle on each of the biconnected components of that vertex. Excluding these cases, Wilson showed that other
15_puzzle
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 graph
Clique_(graph_theory)
handle decompositions of biconnected components. For almost trees of bounded degree (graphs where each biconnected component has at most a constant number
Maximum_common_edge_subgraph
Graph drawing with vertices on a circle
clusters of vertices within a larger graph drawing, such as its biconnected components, clusters of genes in a gene interaction graph, or natural subgroups
Circular_layout
Design technique for parallel algorithms
roots of a forest of rooted trees, connected components, minimum spanning trees, and biconnected components. However, pointer jumping has also shown to
Pointer_jumping
Algorithmic problem of finding non-crossing drawings
all permutations of cyclic edge-order for planar embeddings of biconnected components. Vertex addition methods work by maintaining a data structure representing
Planarity_testing
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
Size of largest complete graph made by contracting edges of a given graph
treewidth is at most two, which is true if and only if each of its biconnected components is a series–parallel graph. Wagner's theorem, which characterizes
Hadwiger_number
Graph formed by subdivision of triangles
although that name has also been used for the biconnected components of a graph that is not itself biconnected. An Apollonian network is a maximal planar
Apollonian_network
Hierarchical clustering of graph edges
graph. The graphs of branchwidth 2 are the graphs in which each biconnected component is a series–parallel graph; the only minimal forbidden minor is
Branch-decomposition
Graph orientation with one source and sink
has an articulation vertex v separating some biconnected component of G from s and t. If this component contains a vertex with a lower number than v,
Bipolar_orientation
strongly a graph is a partial 2-tree if and only if each of its biconnected components is series–parallel. The control-flow graphs arising in the compilation
Partial_k-tree
1990 book by Gerhard Ringel and Nora Hartsfield
theory and spectral graph theory, connectivity of a graph (or even biconnected components), Hall's marriage theorem, line graphs, interval graphs, and the
Pearls_in_Graph_Theory
Index of articles associated with the same name
in a graph Forest (graph theory), an undirected graph with no cycles Biconnected graph, an undirected graph in which every edge belongs to a cycle Directed
Cyclic_graph
American computer scientist
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
Cycle rank Rank (graph theory) SPQR tree St-connectivity Pixel connectivity Vertex separator Strongly connected component Biconnected graph Bridge v t e
St-connectivity
Gluing graphs at complete subgraphs
instance, the SPQR tree of a biconnected graph is a representation of the graph as a 2-clique-sum of its triconnected components. Clique-sums are important
Clique-sum
Set of graph nodes which separate a given pair of nodes if removed
removal of S from the graph separates a and b into distinct connected components. Consider a grid graph with r rows and c columns; the total number n of
Vertex_separator
Israeli-American computer scientist
connected components, spanning trees, biconnected components, Euler tours in trees and graphs, strong orientation, triconnected components, ear decomposition
Uzi_Vishkin
Characteristic of undirected graphs
graph is defined as the number n − c, where c is the number of connected components of the graph. Equivalently, the rank of a graph is the rank of the oriented
Rank_(graph_theory)
Connectivity measure in graph theory
connected components of G. The tree-depth of an undirected graph has a very similar definition, using undirected connectivity and connected components in place
Cycle_rank
Mathematical graph theorem
MR 2965284 Diks, Krzysztof; Stanczyk, Piotr (2010), "Perfect matching for biconnected cubic graphs in O(n log2 n) time", in van Leeuwen, Jan; Muscholl, Anca;
Petersen's_theorem
conjecture 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
Valdes (1980) used triconnected components for structural analysis of biconnected flow graphs. The triconnected components of the undirected version of a
Program_structure_tree
Mathematical problem of making a structure rigid
to be longer. For double bracing, the rigid solutions correspond to biconnected graphs; for tension bracing, they correspond to strongly connected graphs
Grid_bracing
Fewest graph edges whose removal breaks all cycles
internal vertex of a path appears for the first time in that path. In any biconnected graph with circuit rank r {\displaystyle r} , every open ear decomposition
Cyclomatic_number
Graph representing faces of another graph
and only if its dual graph is 3-vertex-connected. Moreover, a planar biconnected graph has a unique embedding, and therefore also a unique dual, if and
Dual_graph
q_{i}\in \{0,1,...,k\},\forall i\in \{1,2,...,N\}} and that at least one component q i = k {\displaystyle q_{i}=k} Let S N d {\displaystyle S_{N}^{d}} represent
Pixel_connectivity
Graph cycle which does not separate remaining elements
In a biconnected graph of circuit rank less than three (such as a cycle graph or theta graph) every cycle is peripheral, but every biconnected graph
Peripheral_cycle
(Connectivity (graph theory)), Graph Biconnectivity (biconnected graph) and Graph Triconnectivity (Triconnected component) problems demonstrated that for some of the
Explicit_multi-threading
Intersection graph of unit intervals on the real line
An indifference graph has a Hamiltonian cycle if and only if it is biconnected. An indifference graph obeys the reconstruction conjecture: it is uniquely
Indifference_graph
Representation of a graph as a path graph "thickened" by some amount
within a constant factor of each other: bounds of this form are known for biconnected outerplanar graphs and for polyhedral graphs. For 2-connected planar
Pathwidth
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
Graph layout on multiple half-planes
the simultaneous embeddability of two graphs whose intersection is a biconnected or a connected graph", Journal of Discrete Algorithms, 14: 150–172, doi:10
Book_embedding
Assignment of colors to edges of a graph
force 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
Edge_coloring
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