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Lower Bounds for Leaf Rank of Leaf Powers

Svein Høgemo

TL;DR

The paper proves that leaf powers can have exponential leaf rank by constructing an infinite family of leaf powers $\{R_n\}$ with $4n$ vertices whose leaf rank is at least $2^{n-2}$. It leverages radial subtree (RS) models and a rooted directed path (RDP) representation to derive large radius requirements for RS models, forcing exponential growth in the leaf root radius. Consequently, the leaf span of leaf powers is non-polynomial, answering an open question about the possible growth of leaf rank and highlighting a substantial separation between leaf powers and their subclasses. The work also discusses NP-membership of leaf-power recognition and implications for potential upper bounds on leaf rank, leaving open whether a tight $2^{\Theta(n)}$ bound holds.

Abstract

Leaf powers and $k$-leaf powers have been studied for over 20 years, but there are still several aspects of this graph class that are poorly understood. One such aspect is the leaf rank of leaf powers, i.e. the smallest number $k$ such that a graph $G$ is a $k$-leaf power. Computing the leaf rank of leaf powers has proved a hard task, and furthermore, results about the asymptotic growth of the leaf rank as a function of the number of vertices in the graph have been few and far between. We present an infinite family of rooted directed path graphs that are leaf powers, and prove that they have leaf rank exponential in the number of vertices (utilizing a type of subtree model first presented by Rautenbach [Some remarks about leaf roots. Discrete mathematics, 2006]). This answers an open question by Brandstädt et al. [Rooted directed path graphs are leaf powers. Discrete mathematics, 2010].

Lower Bounds for Leaf Rank of Leaf Powers

TL;DR

The paper proves that leaf powers can have exponential leaf rank by constructing an infinite family of leaf powers with vertices whose leaf rank is at least . It leverages radial subtree (RS) models and a rooted directed path (RDP) representation to derive large radius requirements for RS models, forcing exponential growth in the leaf root radius. Consequently, the leaf span of leaf powers is non-polynomial, answering an open question about the possible growth of leaf rank and highlighting a substantial separation between leaf powers and their subclasses. The work also discusses NP-membership of leaf-power recognition and implications for potential upper bounds on leaf rank, leaving open whether a tight bound holds.

Abstract

Leaf powers and -leaf powers have been studied for over 20 years, but there are still several aspects of this graph class that are poorly understood. One such aspect is the leaf rank of leaf powers, i.e. the smallest number such that a graph is a -leaf power. Computing the leaf rank of leaf powers has proved a hard task, and furthermore, results about the asymptotic growth of the leaf rank as a function of the number of vertices in the graph have been few and far between. We present an infinite family of rooted directed path graphs that are leaf powers, and prove that they have leaf rank exponential in the number of vertices (utilizing a type of subtree model first presented by Rautenbach [Some remarks about leaf roots. Discrete mathematics, 2006]). This answers an open question by Brandstädt et al. [Rooted directed path graphs are leaf powers. Discrete mathematics, 2010].
Paper Structure (5 sections, 11 theorems, 9 equations, 5 figures)

This paper contains 5 sections, 11 theorems, 9 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Basic Notions
  3. Construction of $R_n$
  4. The Graph Class $\{R_n \mid n\geq 3\}$ has Exponential Leaf Rank
  5. Conclusion

Key Result

lemma 1

If a graph $G$ admits a $k$-leaf root, then it admits a RS model where $\max_{v\in V(G)}r_v \leq k$.

Figures (5)

  • Figure 1: A Hasse diagram of inclusions between leaf powers and some related graph classes. (Abbreviations: LP=Leaf Powers; RDP=Rooted Directed Path graphs; Co-TT=Co-Threshold Tolerance graphs; RS=Graphs with RS models; NeST=Neighborhood Subtree Tolerance graphs.) All inclusions are strict and all non-inclusions are between incomparable graph classes. For more information, see BHTV22BD93BHMW10GMT84MRT88.
  • Figure 2: A rooted directed path model of $R_n$. The arborescence $T$ is a caterpillar; on the right, the tree has been fattened into a box diagram so we can see all the paths.
  • Figure 3: The median of three nodes in a tree.
  • Figure 4: The branching point $m_i$ in a subtree model of $R_n$, and the three cliques it separates.
  • Figure 5: The purple shadow represents the radial subtree $S_{a_i}$ inside $T$, with center $c_{a_i}$ (the solid purple block). It reaches $m_2$ and $s_i$, but not $m_{i+1}$.

Theorems & Definitions (33)

  • definition 1: Caterpillar
  • definition 2: Connector
  • definition 3: $k$-leaf power, $k$-leaf root, leaf rank
  • definition 4: Leaf power
  • definition 5: Leaf span
  • definition 6: Subtree model
  • definition 7: Cover
  • definition 8: Clique subtree
  • definition 9: Radial Subtree model
  • lemma 1
  • ...and 23 more