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Characterization of Trees with Maximum Security

Alex S. A. Alochukwu, Audace A. V. Dossou-Olory, Fadekemi J. Osaye, Valisoa R. M. Rakotonarivo, Shashank Ravichandran, Sarah J. Selkirk, Hua Wang, Hays Whitlatch

TL;DR

This work investigates the extremal behavior of the rank (protection number) in rooted trees by focusing on the security $R(T)=\sum_{v} R_T(v)$ in proper binary trees. It develops a constructive extremal framework based on the binary power representation of the leaf count, saturated-vertex partitions, and switching lemmata that transform any tree into a maximally secure form, yielding explicit maximum security formulas. The authors identify two maximal families: the binary-power tree $T_{\boldsymbol{L}}$ and the almost-complete trees $F(\ell)$, proving they share the same maximal security and providing an algorithm to construct $F(\ell)$ from binary digits. They further extend the analysis to vertex-root protection questions, establishing root-based bounds for general and $k$-ary trees and clarifying when the root can achieve maximal rank. Overall, the paper contributes a new extremal perspective on the protection-number concept, with potential implications for network security design and related combinatorial structures.

Abstract

The rank (also known as protection number or leaf-height) of a vertex in a rooted tree is the minimum distance between the vertex and any of its leaf descendants. We consider the sum of ranks over all vertices (known as the security) in binary trees, and produce a classification of families of binary trees for which the security is maximized. In addition, extremal results relating to maximum rank among all vertices in families of trees is discussed.

Characterization of Trees with Maximum Security

TL;DR

This work investigates the extremal behavior of the rank (protection number) in rooted trees by focusing on the security in proper binary trees. It develops a constructive extremal framework based on the binary power representation of the leaf count, saturated-vertex partitions, and switching lemmata that transform any tree into a maximally secure form, yielding explicit maximum security formulas. The authors identify two maximal families: the binary-power tree and the almost-complete trees , proving they share the same maximal security and providing an algorithm to construct from binary digits. They further extend the analysis to vertex-root protection questions, establishing root-based bounds for general and -ary trees and clarifying when the root can achieve maximal rank. Overall, the paper contributes a new extremal perspective on the protection-number concept, with potential implications for network security design and related combinatorial structures.

Abstract

The rank (also known as protection number or leaf-height) of a vertex in a rooted tree is the minimum distance between the vertex and any of its leaf descendants. We consider the sum of ranks over all vertices (known as the security) in binary trees, and produce a classification of families of binary trees for which the security is maximized. In addition, extremal results relating to maximum rank among all vertices in families of trees is discussed.

Paper Structure

This paper contains 11 sections, 13 theorems, 35 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Most protected binary trees: From binary representation
  3. Partition of a proper binary tree
  4. "Switching" Lemmata
  5. Proof of Theorem \ref{['theo:tl']}
  6. The maximum security in a proper binary tree
  7. Most protected proper binary trees: Almost complete trees
  8. The most protected vertex/root
  9. General trees
  10. $k$-ary trees
  11. Conclusions and future work

Key Result

Theorem 4

Given a proper binary tree $T$ with $\ell$ leaves, where $\ell$ has a binary power representation of $\boldsymbol{L}$, it holds that $R(T) \leq R(T_{\boldsymbol{L}})$.

Figures (19)

  • Figure 1: A binary tree with the protection number of each vertex marked on the vertex.
  • Figure 2: All four non-isomorphic proper binary trees which achieve the maximum total rank of 8 among all proper binary trees with 7 leaves.
  • Figure 3: A proper binary tree $T_{\boldsymbol{L}}$ on $\ell = 11$ leaves that maximizes $R(T)$.
  • Figure 4: Two different proper binary trees corresponding to $\boldsymbol{M}=(2,2,1,0)$ with security 15 and 14, respectively.
  • Figure 5: The trees $T$ (on the left) and $T^*$ (on the right) from Lemma \ref{['lem:sw1']}.
  • ...and 14 more figures

Theorems & Definitions (34)

  • Definition 1: Protection number
  • Definition 2: Security
  • Definition 3
  • Theorem 4
  • Definition 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 24 more