In graph theory, 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 graph.

本章的目标是一个定理：每个有限的图集合中，都存在两个图使得其中一个是另一个的 子式。 图子式：如果无向图H可以通过图G删除边和 顶点 或收缩边得到，则称H为G的子式（minor）或次图。 边收缩：给定边e，两端点为u,v，收缩边e是指将u,v合并为一个点w，且u,v的所有其他边也都连接到w上。 [拟序] 一个具有自反性和传递性的关系称为拟序. [良拟序] 我们称集合 X 上的拟序 \leq 是良拟序，且 X 的元素称为根据 \leq 良拟序的，如果每个无限序列 x_0,x_1,...\in X 中都存在 …

In this section, we explore the following process: 1) bounding the treewidth of a graph, 2) nding the corresponding tree decomposition for this bounded treewidth, and 3) using dynamic programming on tree decompositions.

树分解，属于分解法（Decomposition Method）的一种，也被称为集团树，连接树和连接树，是将图形映射到相关树（Related Tree）的一种方法。 它的主要特性是可以有效地计算原始图的某些属性（例如，独立多项式）。 图的树分解不是唯一的，也不需要与原始图同构。

A graph with tree width w has a tree decomposition where every vertex is labeled by a set of size w+1; the reason for taking the size of the label set to be w+1 is to ensure that trees have tree width 1, which follows from the observations in Example 5.

Tree decompositions # This module implements tree-decomposition methods. A tree-decomposition of a graph G = (V, E) is a pair (X, T), where X = {X 1, X 2, …, X t} is a family of subsets of V, usually called bags, and T is a tree of order t whose nodes are the subsets X i satisfying the following properties: The union of all sets X i equals V.

position). A tree decomposition of a graph G = (V; E) is a tree of N nodes x1; : : : ; xn, with a set Xi V corresponding to each node xi, such that: Every vertex of G belongs to at lea. one set. For every edge in G, there is a set containing both its. ndpoints. For every vertex v in G, the set of bags containing v induces a connecte.

2011年1月21日 · A tree decomposition is a mapping of a graph into a related tree with desirable properties that allow it to be used to efficiently compute certain properties (e.g., independence polynomial) of the original graph. The tree decomposition of a graph is not unique and need not be isomorphic to the original graph.

Let G be a graph. A tree-decomposition of G is a pair (T, W), where T is a tree and W = (Wt : t ∈ V (T )) is such that. if t, t′, t′′ ∈ V (T ) and t′ lies on the path from t to t′′ in T , then Wt ∩Wt′′ ⊆ Wt′. Exercise 2.1.2 ([10]).

We say that a graph G has an H-decomposition if there exists a set L of subgraphs of G, which are isomorphic to H, such that every edge of G appears in exactly one member of L. Note that in order for G to have an H-decomposition, two necessary conditions must hold.