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Trees with proper thinness 2

Flavia Bonomo-Braberman, Ignacio Maqueda, Nina Pardal

TL;DR

This work characterizes trees with proper thinness $2$ and provides a polynomial-time recognition algorithm. By linking strong ordering with a two-class partition, it shows that a tree has $pthin=2$ exactly when a central path $C_0$ contains all high-degree vertices and every degree-3 vertex off $C_0$ attaches to a low-degree vertex on $C_0$, with a forbidden-subgraph list capturing all obstructions. The results also yield a minimal forbidden-induced-subgraph characterization and a certifying algorithm that either identifies $C_0$ or reports a forbidden subgraph, enabling constructive $pthin=2$ representations. The paper further demonstrates that these structural insights do not straightforwardly generalize to $pthin=3$, via a counterexample $T_A$, highlighting intrinsic limitations in extending the approach to higher thinness.

Abstract

The proper thinness of a graph is an invariant that generalizes the concept of a proper interval graph. Every graph has a numerical value of proper thinness and the graphs with proper thinness~1 are exactly the proper interval graphs. A graph is proper $k$-thin if its vertices can be ordered in such a way that there is a partition of the vertices into $k$ classes satisfying that for each triple of vertices $r < s < t$, such that there is an edge between $r$ and $t$, it is true that if $r$ and $s$ belong to the same class, then there is an edge between $s$ and $t$, and if $s$ and $t$ belong to the same class, then there is an edge between $r$ and $s$. The proper thinness is the smallest value of $k$ such that the graph is proper $k$-thin. In this work we focus on the calculation of proper thinness for trees. We characterize trees of proper thinness~2, both structurally and by their minimal forbidden induced subgraphs. The characterizations obtained lead to a polynomial-time recognition algorithm. We furthermore show why the structural results obtained for trees of proper thinness~2 cannot be straightforwardly generalized to trees of proper thinness~3.

Trees with proper thinness 2

TL;DR

This work characterizes trees with proper thinness and provides a polynomial-time recognition algorithm. By linking strong ordering with a two-class partition, it shows that a tree has exactly when a central path contains all high-degree vertices and every degree-3 vertex off attaches to a low-degree vertex on , with a forbidden-subgraph list capturing all obstructions. The results also yield a minimal forbidden-induced-subgraph characterization and a certifying algorithm that either identifies or reports a forbidden subgraph, enabling constructive representations. The paper further demonstrates that these structural insights do not straightforwardly generalize to , via a counterexample , highlighting intrinsic limitations in extending the approach to higher thinness.

Abstract

The proper thinness of a graph is an invariant that generalizes the concept of a proper interval graph. Every graph has a numerical value of proper thinness and the graphs with proper thinness~1 are exactly the proper interval graphs. A graph is proper -thin if its vertices can be ordered in such a way that there is a partition of the vertices into classes satisfying that for each triple of vertices , such that there is an edge between and , it is true that if and belong to the same class, then there is an edge between and , and if and belong to the same class, then there is an edge between and . The proper thinness is the smallest value of such that the graph is proper -thin. In this work we focus on the calculation of proper thinness for trees. We characterize trees of proper thinness~2, both structurally and by their minimal forbidden induced subgraphs. The characterizations obtained lead to a polynomial-time recognition algorithm. We furthermore show why the structural results obtained for trees of proper thinness~2 cannot be straightforwardly generalized to trees of proper thinness~3.
Paper Structure (15 sections, 26 theorems, 2 equations, 6 figures)

This paper contains 15 sections, 26 theorems, 2 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Definitions and basic properties
  3. Trees with proper thinness 2
  4. Interval and proper interval trees and forests
  5. Structural characterization
  6. Structural characterization theorem
  7. Characterization by minimal forbidden induced subgraphs
  8. Polynomial-time recognition algorithm
  9. Trees with proper thinness 3
  10. Proofs
  11. Structural characterization
  12. Auxiliary results
  13. Proof of the structural characterization theorem
  14. Proof of the characterization by minimal forbidden induced subgraphs theorem
  15. Proof of the proposition about the counterexample $T_A$

Key Result

Theorem 1.1

L-B-interval-AT The minimal forbidden induced subgraphs for the class of interval graphs are: bipartite claw, $n$-net for $n\geq 2$, umbrella, $n$-tent for $n\geq 3$, and $C_n$ for $n\geq 4$ (Figure fig:LekkerkerkerBoland).

Figures (6)

  • Figure 1: A $3$-thin representation of a graph (left) and a proper $3$-thin representation of a graph (right). Vertices are ordered from bottom to top, and classes correspond to vertical lines.
  • Figure 2: Minimal forbidden induced subgraphs for the class of interval graphs.
  • Figure 3: Minimal forbidden induced subgraphs for the class of proper interval graphs
  • Figure 4: Minimal forbidden induced subgraphs of trees of proper thinness 2. The dashed lines represent possibly subdivided edges.
  • Figure 5: The tree $T_A$.
  • ...and 1 more figures

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Corollary 2.1
  • Corollary 2.2
  • Theorem 2.2
  • proof
  • Proposition 2.1
  • Proposition 3.1
  • proof
  • ...and 40 more