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The Behavior of Tree-Width and Path-Width under Graph Operations and Graph Transformations

Frank Gurski, Robin Weishaupt

TL;DR

This work provides a comprehensive survey of how tree-width $tw$ and path-width $pw$ behave under a broad spectrum of unary and binary graph transformations, presenting tight bounds when possible and constructive decompositions in linear time. It consolidates and extends known results, showing, for example, precise width changes for vertex/edge deletions, additions, minors, and powers, as well as for line graphs and various graph products. The paper highlights that many transformations preserve width within small additive factors, while others can cause unbounded increases, underscoring the nuanced relationship between graph structure and width. Overall, the results offer a practical toolkit for predicting and computing width changes under common graph operations, with implications for algorithmic design on bounded-width graph classes.

Abstract

Tree-width and path-width are well-known graph parameters. Many NP-hard graph problems allow polynomial-time solutions, when restricted to graphs of bounded tree-width or bounded path-width. In this work, we study the behavior of tree-width and path-width under various unary and binary graph transformations. Doing so, for considered transformations we provide upper and lower bounds for the tree-width and path-width of the resulting graph in terms of the tree-width and path-width of the initial graphs or argue why such bounds are impossible to specify. Among the studied, unary transformations are vertex addition, vertex deletion, edge addition, edge deletion, subgraphs, vertex identification, edge contraction, edge subdivision, minors, powers of graphs, line graphs, edge complements, local complements, Seidel switching, and Seidel complementation. Among the studied, binary transformations we consider the disjoint union, join, union, substitution, graph product, 1-sum, and corona of two graphs.

The Behavior of Tree-Width and Path-Width under Graph Operations and Graph Transformations

TL;DR

This work provides a comprehensive survey of how tree-width and path-width behave under a broad spectrum of unary and binary graph transformations, presenting tight bounds when possible and constructive decompositions in linear time. It consolidates and extends known results, showing, for example, precise width changes for vertex/edge deletions, additions, minors, and powers, as well as for line graphs and various graph products. The paper highlights that many transformations preserve width within small additive factors, while others can cause unbounded increases, underscoring the nuanced relationship between graph structure and width. Overall, the results offer a practical toolkit for predicting and computing width changes under common graph operations, with implications for algorithmic design on bounded-width graph classes.

Abstract

Tree-width and path-width are well-known graph parameters. Many NP-hard graph problems allow polynomial-time solutions, when restricted to graphs of bounded tree-width or bounded path-width. In this work, we study the behavior of tree-width and path-width under various unary and binary graph transformations. Doing so, for considered transformations we provide upper and lower bounds for the tree-width and path-width of the resulting graph in terms of the tree-width and path-width of the initial graphs or argue why such bounds are impossible to specify. Among the studied, unary transformations are vertex addition, vertex deletion, edge addition, edge deletion, subgraphs, vertex identification, edge contraction, edge subdivision, minors, powers of graphs, line graphs, edge complements, local complements, Seidel switching, and Seidel complementation. Among the studied, binary transformations we consider the disjoint union, join, union, substitution, graph product, 1-sum, and corona of two graphs.
Paper Structure (33 sections, 56 theorems, 58 equations, 3 figures, 3 tables)

This paper contains 33 sections, 56 theorems, 58 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs
  4. Tree-width
  5. Path-width
  6. Graph Parameters and Relations
  7. Unary Graph Operations and Graph Transformations
  8. Vertex Deletion and Vertex Addition
  9. Vertex Deletion
  10. Vertex Addition
  11. Edge Addition and Edge Deletion
  12. Edge Deletion
  13. Edge Addition
  14. Subgraph
  15. Vertex Identification
  16. ...and 18 more sections

Key Result

Lemma 1

Every graph of tree-width at most $k$ has a vertex of degree at most $k$.

Figures (3)

  • Figure 1: A graph (left) and an associated tree-decomposition (right).
  • Figure 2: Special graphs.
  • Figure 3: $G_1$, $G_2$, and $G_1 \cup G_2$ for $m = n = 3$ in Example \ref{['ex-union-grid']}.

Theorems & Definitions (58)

  • Definition 1: Tree-width
  • Lemma 1
  • Proposition 2
  • Proposition 3
  • Lemma 4
  • Lemma 5: Clique containment
  • Lemma 6: Complete bipartite subgraph containment
  • Definition 2: Path-width
  • Lemma 7
  • Proposition 8
  • ...and 48 more