The isoperimetric peak of complete trees

TL;DR

This paper determines the exact and bounded values of the isoperimetric peak for complete trees, resolving open problems and introducing novel compression techniques to improve understanding of graph parameters.

Contribution

It provides exact values and bounds for the isoperimetric peak of complete trees, including the case for q-ary trees with q≥5, using new compression methods.

Findings

Isoperimetric peak equals depth for q≥5 in complete q-ary trees.

Vertex separation number can differ significantly from the isoperimetric peak.

New bounds on pathwidth and pursuit-evasion parameters for complete trees.

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.