[Paper Review] On homometric sets in graphs

Maria Axenovich, Lale Özkahya|arXiv (Cornell University)|Mar 6, 2012

Graph Labeling and Dimension Problems3 citations

TL;DR

This paper improves bounds on the maximum size of disjoint homometric vertex sets in graphs, introducing new lower bounds for trees and graphs of bounded diameter. It proves that for any tree T on n vertices, h(T) ≥ n^{1/3} − 1, and for graphs of diameter d, h(G) ≥ c n^{1/(2d−2)}, significantly advancing prior results on the homometric set problem in graph theory.

ABSTRACT

For a vertex set $S\subseteq V(G)$ in a graph $G$, the {\em distance multiset}, $D(S)$, is the multiset of pairwise distances between vertices of $S$ in $G$. Two vertex sets are called {\em homometric} if their distance multisets are identical. For a graph $G$, the largest integer $h$, such that there are two disjoint homometric sets of order $h$ in $G$, is denoted by $h(G)$. We slightly improve the general bound on this parameter introduced by Albertson, Pach and Young (2010) and investigate it in more detail for trees and graphs of bounded diameter. In particular, we show that for any tree $T$ on $n$ vertices $h(T) \geq \sqrt[3]{n}$ and for any graph $G$ of fixed diameter $d$, $h(G) \geq cn^{1/ (2d-2)}$.

Motivation & Objective

• To improve the general upper and lower bounds on the maximum size h(G) of two disjoint homometric vertex sets in a graph G.
• To investigate h(G) for specific graph classes, particularly trees, spiders, caterpillars, and haircombs.
• To establish quantitative relationships between h(G) and graph parameters such as diameter, edge count, and vertex degree distribution.
• To extend known results from Euclidean distance sets and integer homometric sets to graph-theoretic settings.

Proposed method

• Introduces a novel construction of homometric sets using structural decomposition of trees, particularly focusing on spiders, caterpillars, and haircombs.
• Applies number-theoretic and graph-theoretic tools, including multiset differences and isomorphism-based constructions, to generate homometric pairs.
• Employs degree-based vertex partitioning and distance-based partitioning (e.g., by distance from a leaf) to derive lower bounds on h(T).
• Uses the concept of 'bad' vertices (degree ≥ 3) and their distribution to bound the size of homometric sets via structural lemmas.
• Applies extremal graph theory techniques to derive bounds in terms of diameter d and edge count e, using inequalities involving binomial coefficients.
• Leverages known results on homometric sets in integer multisets (e.g., U+V and U−V constructions) to guide constructions in graphs.

Experimental results

Research questions

• RQ1 What is the best possible lower bound on h(G) for trees in terms of n, the number of vertices?
• RQ2 How does the diameter d of a graph influence the maximum size of disjoint homometric sets?
• RQ3 Can improved lower bounds on h(G) be derived using edge density and structural parameters like vertex degrees?
• RQ4 For which classes of trees—such as spiders, caterpillars, or haircombs—is h(T) bounded below by a nontrivial function of n?
• RQ5 Are there graph families where the standard homometric set construction fails, and if so, how can such cases be characterized?

Key findings

• For any tree T on n vertices, h(T) ≥ n^{1/3} − 1, significantly improving previous bounds.
• For graphs of fixed diameter d, h(G) ≥ c n^{1/(2d−2)} for some constant c > 0, with the bound holding for sufficiently large n.
• For caterpillars, h(R_n) ≥ n/6; for haircombs, h(H_n) ≥ √n / 2.
• For n-vertex spiders with k legs, h(S_{n,k}) ≥ (1/4 + 3/(8k−12))n when k ≥ 5, and exact values are given for k = 3 and k = 4.
• For spiders with n/2 legs, h(S_{n,n/2}) = (n+2)/4, and h(T) = n/2 for certain symmetric 3-legged spiders.
• When the number of 'bad' vertices (degree ≥ 3) is o(√n), or when the number of leaves is large, h(T) = Ω(√n), even for trees with small diameter.