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Number denoting a graph's closeness to a tree
the treewidth of an undirected graph is an integer number which specifies, informally, how far the graph is from being a tree. The smallest treewidth is
Treewidth
Graph where all long cycles have a chord
coloring may be solved in polynomial time when the input is chordal. The treewidth of an arbitrary graph may be characterized by the size of the cliques
Chordal_graph
Mapping of a graph into a tree
decomposition is a mapping of a graph into a tree that can be used to define the treewidth of the graph and speed up solving certain computational problems on the
Tree_decomposition
On linear-time algorithms for graph logic
second-order logic of graphs can be decided in linear time on graphs of bounded treewidth. The result was first proved by Bruno Courcelle in 1990 and independently
Courcelle's_theorem
Measure of graph complexity
structural complexity of the graph; it is closely related to treewidth, but unlike treewidth it can be small for dense graphs. It is defined as the minimum
Clique-width
Numerical invariant of graphs
graphs and the star height of regular languages. Intuitively, where the treewidth of a graph measures how far it is from being a tree, this parameter measures
Tree-depth
Hierarchical clustering of graph edges
And as with treewidth, many graph optimization problems may be solved efficiently for graphs of small branchwidth. However, unlike treewidth, the branchwidth
Branch-decomposition
Probabilistic graphical representation of causal relationships
bounded treewidth is necessary to allow exact, tractable inference, since the worst-case inference complexity is exponential in the treewidth k (under
Bayesian_network
of graph, defined either as a subgraph of a k-tree or as a graph with treewidth at most k. Many NP-hard combinatorial problems on graphs are solvable
Partial_k-tree
Graph which can be made planar by removing a single node
embedding, Hadwiger's conjecture, YΔY-reducible graphs, and relations between treewidth and graph diameter. Apex graphs are closed under the operation of taking
Apex_graph
Method of graph decomposition
with each of the subgraphs. Brambles may be used to characterize the treewidth of G. A haven of order k in a graph G is a function β that maps each set
Bramble_(graph_theory)
Representation of a graph as a path graph "thickened" by some amount
searching number. Pathwidth and path-decompositions are closely analogous to treewidth and tree decompositions. They play a key role in the theory of graph minors:
Pathwidth
Method of graph decomposition
introduced by Seymour & Thomas (1993) as a tool for characterizing the treewidth of graphs. Their other applications include proving the existence of small
Haven_(graph_theory)
It is NP-complete but fixed-parameter tractable on graphs of bounded treewidth. The monochromatic triangle problem takes as input an n-node undirected
Monochromatic_triangle
vertices in any of its bags; the treewidth of G is the minimum width of any tree decomposition of G. treewidth The treewidth of a graph G is the minimum width
Glossary_of_graph_theory
whose treewidth is bounded by a constant can be performed in polynomial time. Here, the treewidth can be the primal treewidth, dual treewidth, or incidence
♯SAT
Subgraph with contracted edges
per subgraph. Even stronger, for any fixed H, H-minor-free graphs have treewidth O ( n ) {\displaystyle \scriptstyle O({\sqrt {n}})} . The Hadwiger conjecture
Graph_minor
Theorem about infinite graphs
Halin (1965), and is a precursor to the work of Robertson and Seymour linking treewidth to large grid minors, which became an important component of the algorithmic
Halin's_grid_theorem
Mathematical tree with cycle through leaves
graphs can be recognized in linear time. Because Halin graphs have low treewidth, many computational problems that are hard on other kinds of planar graphs
Halin_graph
Type of planar graph
{\displaystyle k} -outerplanar graphs have treewidth at most 3 k − 1 {\displaystyle 3k-1} . However, some bounded-treewidth planar graphs such as the nested triangles
K-outerplanar_graph
Problem in graph theory
be found in polynomial time in graphs of bounded treewidth. That is, when parameterized by treewidth rather than by the cut size, the weighted maximum
Maximum_cut
Form of second-order logic
algorithms for evaluating monadic second-order formulas over graphs of bounded treewidth. It is also of fundamental importance in automata theory, where the
Monadic_second-order_logic
Graph theory model
the maximal graphs with a treewidth of k ("maximal" means that no more edges can be added without increasing their treewidth). They are also exactly the
K-tree
Graph that can be embedded in the plane
Halin graph is planar. Like outerplanar graphs, Halin graphs have low treewidth, making many algorithmic problems on them more easily solved than in unrestricted
Planar_graph
Logical formulation of graph properties
must have bounded treewidth. The proof is based on a theorem of Robertson and Seymour that the families of graphs with unbounded treewidth have arbitrarily
Logic_of_graphs
Chordal graph with the given graph as a subgraph
maximum clique in the resulting chordal graph, can be used to define the treewidth of G. Chordal completions can also be used to characterize several other
Chordal_completion
Recursively-formed graph with two terminal vertices
edge. Every series–parallel graph has treewidth at most 2 and branchwidth at most 2. Indeed, a graph has treewidth at most 2 if and only if it has branchwidth
Series–parallel_graph
Graph formed by subdivision of triangles
decomposition into three trees. They are the maximal planar graphs with treewidth three, a class of graphs that can be characterized by their forbidden
Apollonian_network
Non-crossing graph with vertices on outer face
cycle. Every outerplanar graph is 3-colorable, and has degeneracy and treewidth at most 2. The outerplanar graphs are a subset of the planar graphs, the
Outerplanar_graph
Problem of finding the longest simple path for a given graph
path problem is also fixed-parameter tractable when parameterized by the treewidth of the graph. For graphs of bounded clique-width, the longest path can
Longest_path_problem
Describing a family of graphs by excluding certain (sub)graphs
Bodlaender, Hans L. (1998), "A partial k-arboretum of graphs with bounded treewidth", Theoretical Computer Science, 209 (1–2): 1–45, doi:10.1016/S0304-3975(97)00228-4
Forbidden graph characterization
Forbidden_graph_characterization
in polynomial time for graphs of bounded treewidth, but the exponent of the polynomial depends on the treewidth. It may be solved in linear time for precoloring
Precoloring_extension
Structure-preserving correspondence between node-link graphs
The crucial property turns out to be treewidth, a measure of how tree-like the graph is. For a graph G of treewidth at most k and a graph H, the homomorphism
Graph_homomorphism
Measurement of graph sparsity
equal to the treewidth and at most equal to the pathwidth. However, there exist graphs with bounded degeneracy and unbounded treewidth, such as the grid
Degeneracy_(graph_theory)
Any planar graph can be subdivided by removing a few vertices
a class of graphs can be formalized and quantified by the concepts of treewidth and polynomial expansion. As it is usually stated, the separator theorem
Planar_separator_theorem
Italian mathematician (1824–1897)
formula Brioschi normal form Hermite–Kronecker–Brioschi characterization Treewidth Scientific career Fields Mathematics Institutions University of Pavia
Francesco_Brioschi
Order-zero graph or any edgeless graph
∞ Automorphisms 1 Chromatic number 0 Chromatic index 0 Genus 0 Properties Integral Symmetric Treewidth -1 Notation K0 Table of graphs and parameters
Null_graph
Gluing graphs at complete subgraphs
the operation. Clique-sums have a close connection with treewidth: If two graphs have treewidth at most k, so does their k-clique-sum. Every tree is the
Clique-sum
Number of vertices with unambiguous distances
a minor and also gave bounds for chordal graphs and graphs of bounded treewidth. The authors Foucaud et al. (2017a) proved bounds of the form n = O (
Metric dimension (graph theory)
Metric_dimension_(graph_theory)
Unproven computational hardness assumption
parameterized complexity of several graph problems on graphs of bounded treewidth. In particular, if the strong exponential time hypothesis is true, then
Exponential_time_hypothesis
Graph with at most one crossing per edge
have bounded local treewidth, meaning that there is a (linear) function f such that the 1-planar graphs of diameter d have treewidth at most f(d); the
1-planar_graph
Assignment of colors to edges of a graph
particular, Zhou, Nakano & Nishizeki (1996) showed that for graphs of treewidth w, an optimal edge coloring can be computed in time O(nw(6w)w(w + 1)/2)
Edge_coloring
Graph with a prism as its skeleton
the graphs of treewidth three. The triangular prism and cube graph have treewidth exactly three, but all larger prism graphs have treewidth four. Other
Prism_graph
bidimensionality are based on the following combinatorial property: either the treewidth of a graph is small, or the graph contains a large grid as a minor or
Bidimensionality
Undirected graph with 11 nodes and 27 edges
{\displaystyle k} -tree, it has treewidth 3, and its graph is maximal, meaning it can add no more edges without increasing its treewidth. Both of its maximal cliques
Goldner–Harary_graph
Node-weighted undirected graph associated with a given combinatorial optimization problem
constraint satisfaction problem can be solved in time exponential only in the treewidth of its variable-interaction graph (constraint network). However, a major
Constraint_composite_graph
Method for finding patterns in networks
algorithms when the subgraph pattern that it is trying to detect has bounded treewidth. The color-coding method was proposed and analyzed in 1994 by Noga Alon
Color-coding
Inference algorithm for probabilistic graphical models
exponential time complexity, but could be efficient in practice for low-treewidth graphs, if the proper elimination order is used. Enabling a key reduction
Variable_elimination
Edges that hit all cycles in a graph
much easier to compute than the minimum feedback arc set. For graphs of treewidth t {\displaystyle t} , dynamic programming on a tree decomposition of the
Feedback_arc_set
Cubic graph with 8 vertices and 12 edges
graph is also one of four minimal forbidden minors for the graphs of treewidth at most three (the other three being the complete graph K5, the graph
Wagner_graph
Number of cops needed to catch a robber on a graph
that the cop number is generally smaller than the treewidth. More specifically, on graphs of treewidth w {\displaystyle w} , the cop number is at most ⌊
Cop_number
Machine learning algorithm
propagation) Note that this last step is inefficient for graphs of large treewidth. Computing the messages to pass between supernodes involves doing exact
Junction_tree_algorithm
Finiteness of sets of forbidden graph minors
Verdière graph invariant bounded by some fixed constant; graphs with treewidth, pathwidth, or branchwidth bounded by some fixed constant. Some examples
Robertson–Seymour_theorem
Computer science field
restrictions on the input structure: for instance, requiring that it has treewidth bounded by a constant (which more generally implies the tractability of
Model_checking
Maximum number of colors obtainable by a greedy graph coloring algorithm
parameterized by both the treewidth and the Grundy number, although (assuming the exponential time hypothesis) the dependence on treewidth must be greater than
Grundy_number
Unsolved problem in computational complexity theory
Permutation graphs Circulant graphs Bounded-parameter graphs Graphs of bounded treewidth Graphs of bounded genus (Planar graphs are graphs of genus 0.) Graphs
Graph_isomorphism_problem
On graph drawing with integer edge lengths
bipartite planar graphs, (2,1)-sparse planar graphs, planar graphs of treewidth at most 3, and graphs of degree at most four that either contain a diamond
Harborth's_conjecture
Capital of Sulu province, Philippines
H&startDocNo=1&resultsUrlKey=29_T5666963374&cisb=22_T5667573664&treeMax=true&treeWidth=0&csi=173384&docNo=2[full citation needed] "The Official Website of the
Jolo,_Sulu
number two testing Recognizing string graphs Subgraph isomorphism problem Treewidth Testing whether a tree may be represented as Euclidean minimum spanning
List_of_NP-complete_problems
In applied mathematics, a technique to find the shortest path
$(1+\varepsilon)$-Embedding of Low Highway Dimension Graphs into Bounded Treewidth Graphs". SIAM Journal on Computing. 47 (4): 1667–1704. arXiv:1502.04588
Contraction_hierarchies
Computational hardness assumption
problem of approximating the treewidth of graphs, a structural parameter closely related to expansion. For graphs of treewidth w {\displaystyle w} , the
Small set expansion hypothesis
Small_set_expansion_hypothesis
Generalization of depth-first search trees
involving orientations to be recognized efficiently for graphs of bounded treewidth using Courcelle's theorem. Not every infinite connected graph has a Trémaux
Trémaux_tree
Bodlaender, Hans L. (1988), "Dynamic programming on graphs with bounded treewidth", in Lepistö, Timo; Salomaa, Arto (eds.), Automata, Languages and Programming
Baker's_technique
Set of objects whose state must satisfy limits
be tractable are those where the hypergraph of constraints has bounded treewidth, or where the constraints have arbitrary form but there exist equationally
Constraint satisfaction problem
Constraint_satisfaction_problem
Invariant in graph theory
whether the queue number of a graph could be bounded as a function of its treewidth, and cited an unpublished Ph.D. dissertation of S. V. Pemmaraju as providing
Queue_number
Intersection graph of a chord diagram
restricted to circle graphs. For instance, Kloks (1996) showed that the treewidth of a circle graph can be determined, and an optimal tree decomposition
Circle_graph
Graph coloring variant in graph theory
question. However, it can be solved in polynomial time for graphs of bounded treewidth and bounded diameter. For path graphs P n {\displaystyle P_{n}} : χ ρ
Packing_coloring
Tree graph with all nodes within distance 1 from central path
parametrized algorithm that finds an optimal solution for the MSCP in bounded treewidth graphs. So both the Spanning Caterpillar Problem and the MSCP have linear
Caterpillar_tree
may also be solved exactly, in polynomial time, for graphs of bounded treewidth. Koh & Tay (2002). Schrijver (1983). Welsh (1997). Noy (2001). Las Vergnas
Strong_orientation
Canadian computer scientist and mathematician
drawing, for the structural theory of graph width parameters including treewidth and queue number, and for the use of these parameters in the parameterized
Vida_Dujmović
Hypothesis in computational complexity theory
Austrin, Per; Pitassi, Toniann; Liu, David (2014). "Inapproximability of Treewidth and Related Problems". Journal of Artificial Intelligence Research. 49:
Computational hardness assumption
Computational_hardness_assumption
Economical computational problem
present algorithms for graphical games where the graphs with bounded treewidth. They show that a PNE can be found in polynomial time using tree decomposition
Nash_equilibrium_computation
Dutch computer scientist
Michael Fellows, and Danny Hermelin on kernelization. A festschrift, Treewidth, Kernels, and Algorithms: Essays Dedicated to Hans L. Bodlaender on the
Hans_L._Bodlaender
Graph layout on multiple half-planes
not known until 2020, when Bekos et al. presented planar graphs with treewidth 4 that require four pages in any book embedding. Subdividing every edge
Book_embedding
Property in graph theory
the treewidth or pathwidth of the same graph. However, it is at most the pathwidth multiplied by O ( Δ ) {\displaystyle O(\Delta )} , or the treewidth multiplied
Cutwidth
Problem on triangles in graph theory
have degeneracy at most five.) It is also known to hold for graphs of treewidth at most six, for threshold graphs, for sufficiently dense graphs, and
Tuza's_conjecture
Longest distance between two vertices
time for interval graphs, and in near-linear time for graphs of bounded treewidth. In median graphs, the diameter can be found in the subquadratic time
Diameter_(graph_theory)
Graph width parameter
Because treewidth and branchwidth are always within constant factors of each other, similar bounds can be used to relate carving width to treewidth. Another
Carving_width
Graph representing a permutation
independent set of the same size in the corresponding permutation graph. the treewidth and pathwidth of permutation graphs can be computed in polynomial time;
Permutation_graph
D-separation Markov random field Tree decomposition (Junction tree) and treewidth Graph triangulation (see also Chordal graph) Perfect order Hidden Markov
List_of_graph_theory_topics
1997 studio album by Gary Barlow
cNo=1&resultsUrlKey=29_T12772348940&cisb=22_T12772348939&treeMax=true&treeWidth=0&csi=172244&docNo=2 Nexis Lexis, Search Gary Barlow Open Road, 2nd source
Open_Road_(Gary_Barlow_album)
Graph data structure
reduction from the set cover problem. However, for graphs with bounded treewidth, there is a linear-time, fixed-parameter tractable algorithm. An e-graph
E-graph
Upper bound on intersecting set families
MR 4135291, S2CID 57761161 Harvey, Daniel J.; Wood, David R. (2014), "Treewidth of the Kneser graph and the Erdős–Ko–Rado theorem", Electronic Journal
Erdős–Ko–Rado_theorem
Node labeling problem in graph theory
1016/0012-365X(75)90039-4. MR 0427150. Gruber, Hermann (2012). "On Balanced Separators, Treewidth, and Cycle Rank". Journal of Combinatorics. 3 (4): 669–682. arXiv:1012
Graph_bandwidth
dimension parameter is incomparable to structural graph parameters such as treewidth, cliquewidth, or minor-freeness. On the other hand, a star with unit edge
Highway_dimension
Category of routing problem minimizing total distance and time
|V|^{3})} -time algorithm In FPT with respect to |A| In XP with respect to treewidth Windy NP-complete P in some special cases Factor 3/2 k-Hierarchical NP-complete
Arc_routing
Metric space with double measurement
authors list (link) Zhou, Felix (21 Feb 2023). "Doubling Dimension and Treewidth" (PDF). Pansu, Pierre (1989). "Métriques de Carnot-Carathéodory et quasiisométries
Doubling_space
Australian mathematician
in graph product structure theory, extremal graph minor theory, graph treewidth, graphs on surfaces, graph colouring, geometric graph theory, poset dimension
David_Wood_(mathematician)
Type of algorithm
general Steiner Forest problem is NP-hard on graphs of treewidth 3. However, on graphs of treewidth t an EPAS can compute a ( 1 + ε ) {\displaystyle (1+\varepsilon
Parameterized approximation algorithm
Parameterized_approximation_algorithm
Mathematical game/problem
move turn by turn, but move simultaneously) is equivalent to finding the treewidth of G, and a winning strategy for the evader may be described in terms
Pursuit–evasion
British mathematician
separator theorem of Richard Lipton and Robert Tarjan; a paper characterizing treewidth in terms of brambles; and a polynomial-time algorithm to compute the branch-width
Paul_Seymour_(mathematician)
Graph coloring with equal color classes
whether G admits an equitable c-coloring in time O(nO(t)), where t is the treewidth of G; in particular, equitable coloring may be solved optimally in polynomial
Equitable_coloring
Size of largest complete graph made by contracting edges of a given graph
cycle 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
algorithms for Treewidth and related graph problems A C++ implementation used in the paper "A complete Anytime Algorithm for Treewidth, Vibhav Gogate
Decomposition method (constraint satisfaction)
Decomposition_method_(constraint_satisfaction)
Binary operation in graph theory
planar graph is a subgraph of a strong product of a path and a graph of treewidth at most six. This result has been used to prove that planar graphs have
Strong_product_of_graphs
Algorithmically defined graph
with 2 log2 n + O(log log n) bits per label, and that graphs of bounded treewidth have a labeling scheme with log2 n + O(log log n) bits per label. The
Implicit_graph
Israeli mathematician and computer scientist
polynomial relation between the size of a grid graph minor of a graph and its treewidth.[CC16] This connection between these two graph properties is a key component
Julia_Chuzhoy
finite k, the monadic second-order (MSO) theory of countable graphs with treewidth ≤k (and a corresponding tree decomposition) is interpretable in S2S (see
S2S_(mathematics)
Type of database query
in which all relations used are binary, this notion corresponds to the treewidth of the dependency graph of the variables in the query (i.e., the graph
Conjunctive_query
DocNo=1&resultsUrlKey=29_T2284942163&cisb=22_T2284942162&treeMax=true&treeWidth=0&csi=10882&docNo=3> accessed at Lexis Nexis Academic. "What is the difference
Pediatric_plastic_surgery
TREEWIDTH
TREEWIDTH
TREEWIDTH
TREEWIDTH
Boy/Male
Polish
Born in January.
Boy/Male
Arabic, Muslim
Distant
Girl/Female
Arabic, Muslim
Music of Night
Surname or Lastname
English and Irish
English and Irish : variant of Diamond 2 and 3.
Boy/Male
Indian
Quarrdsome
Boy/Male
Hindu, Indian, Sanskrit
Bestower of Dharma
Girl/Female
Swedish Teutonic German
Warrior maid.
Boy/Male
Tamil
Sreekumar | à®·à¯à®°à¯€à®•à¯à®®à®¾à®°
Wealthy person
Female
English
 Latin form of Greek Chloē, CHLOE means "green shoot." In mythology, this is a surname of the goddess Demeter. In the New Testament bible, this name is mentioned by Paul in 1 Corinthians 1:11.
Boy/Male
Arabic
Wishes
TREEWIDTH
TREEWIDTH
TREEWIDTH
TREEWIDTH
TREEWIDTH