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Thick Forests

Martin Dyer, Haiko Müller

TL;DR

We introduce thick graphs as a framework in which a graph G is represented by thick vertices (cliques) and thick edges (cobipartite links) with a thin graph H that captures the skeleton. Focusing on thick forests, we show these graphs form a large class of perfect graphs and provide a polynomial-time recognition algorithm, contrasting with NP-complete recognition in general thick graph classes. We study two classic counting problems—independent sets and colorings—developing exact polynomial-time counting for independent sets in a broad class Q of quasi thick forests and polynomial-time approximation schemes for colorings via clique-cutset decompositions; exact counting of colorings remains #P-complete even for thick forests, but approximation is tractable in Q. Finally, we examine parameterisations by the thin graph size and by treewidth, yielding XP and, in some cases, fixed-parameter tractable results for counting independent sets and related tasks, and highlighting the boundary between tractable and intractable thick-graph families.

Abstract

We consider classes of graphs, which we call thick graphs, that have the vertices of a corresponding thin graph replaced by cliques and the edges replaced by cobipartite graphs In particular, we consider the case of thick forests, which we show to be the largest class of perfect thick graphs. Recognising membership of a class of thick graphs is NP-complete unless the class of thin graphs is triangle-free, so we focus on this case. Even then membership can be NP-complete. However, we show that the class of thick forests can be recognised in polynomial time. We consider two well-studied combinatorial problems on thick graphs, independent sets and proper colourings. Since determining the independence or chromatic number of a perfect graph is known to be tractable, we examine the complexity of counting all independent sets and colourings in thick forests. Finally, we consider two parametric extensions to larger classes of thick graphs: where the parameter is the size of the thin graph, and where the parameter is its treewidth.

Thick Forests

TL;DR

We introduce thick graphs as a framework in which a graph G is represented by thick vertices (cliques) and thick edges (cobipartite links) with a thin graph H that captures the skeleton. Focusing on thick forests, we show these graphs form a large class of perfect graphs and provide a polynomial-time recognition algorithm, contrasting with NP-complete recognition in general thick graph classes. We study two classic counting problems—independent sets and colorings—developing exact polynomial-time counting for independent sets in a broad class Q of quasi thick forests and polynomial-time approximation schemes for colorings via clique-cutset decompositions; exact counting of colorings remains #P-complete even for thick forests, but approximation is tractable in Q. Finally, we examine parameterisations by the thin graph size and by treewidth, yielding XP and, in some cases, fixed-parameter tractable results for counting independent sets and related tasks, and highlighting the boundary between tractable and intractable thick-graph families.

Abstract

We consider classes of graphs, which we call thick graphs, that have the vertices of a corresponding thin graph replaced by cliques and the edges replaced by cobipartite graphs In particular, we consider the case of thick forests, which we show to be the largest class of perfect thick graphs. Recognising membership of a class of thick graphs is NP-complete unless the class of thin graphs is triangle-free, so we focus on this case. Even then membership can be NP-complete. However, we show that the class of thick forests can be recognised in polynomial time. We consider two well-studied combinatorial problems on thick graphs, independent sets and proper colourings. Since determining the independence or chromatic number of a perfect graph is known to be tractable, we examine the complexity of counting all independent sets and colourings in thick forests. Finally, we consider two parametric extensions to larger classes of thick graphs: where the parameter is the size of the thin graph, and where the parameter is its treewidth.
Paper Structure (27 sections, 36 theorems, 4 equations, 19 figures)

This paper contains 27 sections, 36 theorems, 4 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Previous work
  3. Preliminaries
  4. Graph classes
  5. Treewidth
  6. Thick graphs
  7. Full links and full graphs
  8. Subcolouring
  9. Altering the model
  10. Thick forests
  11. Thick bipartite graphs
  12. Forbidden subgraphs
  13. Recognising a thick forest
  14. Recognising a cobipartite graph
  15. Recognising a unipolar graph
  16. ...and 12 more sections

Key Result

Lemma 1

Let $\mathscr{C}$ be any class such that $\mathscr{C}\nsubseteq\mathscr{T}$, then the recognition problem for $\textup{\sf thick}(\mathscr{C})\xspace$ is NP-complete.

Figures (19)

  • Figure 1: Graph classes and their inclusions
  • Figure 2: Loose triangles and 4-cycles
  • Figure 3: Full links
  • Figure 4: $C_4$, a thick $C_4$ and a full $C_4$
  • Figure 5: A thick triangle with a 3-subcolouring
  • ...and 14 more figures

Theorems & Definitions (65)

  • Lemma 1
  • proof
  • Lemma 2: Dyer, Greenhill, Müller
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Corollary 1
  • Lemma 5
  • ...and 55 more