The isoperimetric peak of complete trees

TL;DR

The paper tackles the vertex-isoperimetric peak $\Phi(G)$ for complete $q$-ary trees of depth $d$, introducing a three-stage compression framework—left, push-down, and aeolian compressions—that canonizes isoperimetrically optimal vertex subsets. Using these tools, it proves exact values for $q\ge 5$ ($\Phi_V(T)=d$) and near-exact bounds for $q\in\{3,4\}$, while giving pathwidth-based upper bounds that bound $\Phi_V(T)$ for all $q$, including $q=2$. The results shed light on the shape of isoperimetrically optimal sets on trees (often disconnected and depth-first flavored) and link the isoperimetric parameter to pathwidth and vertex-separation, yielding new bounds and separations between these graph parameters. Applications demonstrate that vertex separation can be arbitrarily far from the isoperimetric peak and provide improved bounds for related pursuit-evasion and visibility parameters. The work advances the understanding of isoperimetric structure on trees and offers a versatile compression toolkit for exact and approximate analyses.

Abstract

We give exact values and bounds on the isoperimetric peak of complete trees, improving on known results. For the complete $q$-ary tree of depth $d$, if $q\ge 5$, then we find that the isoperimetric peak equals $d$, completing an open problem. In the case that $q$ is 3 or 4, we determine the value up to three values, and in the case $q=2$, up to a logarithmic additive factor. Our proofs use novel compression techniques, including left, down, and aeolian compressions. We apply our results to show that the vertex separation number and the isoperimetric peak of a graph may be arbitrarily far apart as a function of the order of the graph and give new bounds on the pathwidth and pursuit-evasion parameters on complete trees.

Figures (7)

• Figure 1: An example of a $1$-compression and then a $2$-compression on a subset of vertices in the square grid, where the subset of vertices are the four vertices represented by filled vertices.
• Figure 2: An example of the action of the left-compression on a subset of vertices in a complete binary tree.
• Figure 3: An example of a treeswap of $S$ between vertices $u$ and $v$, where $S$ is the set of filled vertices.
• Figure 4: The left-compressed subset of vertices in a complete binary tree that has one vertex in level 1, two vertices in levels 2 and 3, and nine vertices in level 4.
• Figure 5: The results of performing a $u_1$--$v_1$ exchange and a $u_2$--$v_2$ exchange on a subset of vertices $S$ (left), which yields a down-compressed set $S'$ (right)

Theorems & Definitions (30)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Lemma 7
• proof
• Lemma 8
• proof