Math 2.0

Combinatorics Definition

Treewidth

Builds on GraphTree

Definition

Let G=(V,E)G = (V, E) be a finite graph. A tree decomposition of GG is a pair (T,(Xt)t∈V(T))(T, (X_t)_{t \in V(T)}) with the following properties. Here TT is a tree, and each XtX_t is a subset of VV, called a bag.

  1. Every vertex of GG lies in some bag.
  2. For every edge uvuv of GG, some bag contains both uu and vv.
  3. For every vertex vv of GG, the nodes tt with v∈Xtv \in X_t form a connected subtree of TT.

The width of the decomposition is max⁡t∣Xt∣−1\max_t \abs{X_t} - 1. The treewidth tw⁡(G)\operatorname{tw}(G) is the least width of a tree decomposition of GG.

Remarks

Example: the cycle C4C_4 with vertices 1,2,3,41, 2, 3, 4 in order has the tree decomposition with two adjacent nodes and bags {1,2,3}\set{1, 2, 3} and {1,3,4}\set{1, 3, 4}. Each vertex is in a bag, each edge lies in a bag, and the nodes containing 11 or 33 form the whole tree, so the width is 22. Non-example: put the bags {1,2},{2,3},{3,4},{4,1}\set{1, 2}, \set{2, 3}, \set{3, 4}, \set{4, 1} in this order along a path. They satisfy clauses 1 and 2 but fail clause 3, because vertex 11 lies in the first and last bags only. Every clique of GG lies in a single bag, so tw⁡(Kn)=n−1\operatorname{tw}(K_n) = n - 1. A graph with at least one edge has treewidth 11 exactly when it is a forest, and the k×kk \times k grid with k≥2k \ge 2 has treewidth kk.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

Show the path as a graph
Graph, definition, no evidence levelGraphdefinitionWalk, definition, no evidence levelWalkdefinitionReachability, definition, no evidence levelReachabilitydefinitionTrail, definition, no evidence levelTraildefinitionConnected graph, definition, no evidence levelConnected graphdefinitionCircuit, definition, no evidence levelCircuitdefinitionCycle, definition, no evidence levelCycledefinitionAcyclic graph, definition, no evidence levelAcyclic graphdefinitionTree, definition, no evidence levelTreedefinitionTreewidth, definition, no evidence levelTreewidthdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Graph definition
  2. Walk definition
  3. Reachability definition
  4. Trail definition
  5. Connected graph definition
  6. Circuit definition
  7. Cycle definition
  8. Acyclic graph definition
  9. Tree definition
  10. Treewidth definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Treewidth
About this page
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Kinddefinition
WritingW0 stub
MSC05C83, 05C75, 05C85
DefinesTreewidth
UsesGraph, Tree
Changed2 November 2026

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