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On Separating Path and Tree Systems in Graphs

Ahmad Biniaz, Prosenjit Bose, Jean-Lou De Carufel, Anil Maheshwari, Babak Miraftab, Saeed Odak, Michiel Smid, Shakhar Smorodinsky, Yelena Yuditsky

TL;DR

This work studies vertex-separating path and tree systems in graphs, formalizing the parameters $f(G)$ and $f_t(G)$ and establishing tight bounds across key graph classes. The authors develop constructive labeling and decomposition techniques, deriving sharp results for complete bipartite graphs, grids, and maximal outerplanar graphs, and they prove a fundamental lower bound $f(T)\ge \frac{n}{4}$ for trees with matching constructions. They further advance the theory by linking vertex-separating path systems to VC-dimension concepts to obtain polynomial lower bounds for certain minor-free graph classes, and by providing radius-based bounds for separating tree systems. The results deepen understanding of how efficiently vertex separation can be achieved with paths or trees and open several algorithmic and structural questions for broader graph families, including triangulations and graph products.

Abstract

We explore the concept of separating systems of vertex sets of graphs. A separating system of a set $X$ is a collection of subsets of $X$ such that for any pair of distinct elements in $X$, there exists a set in the separating system that contains exactly one of the two elements. A separating system of the vertex set of a graph $G$ is called a vertex-separating path (tree) system of $G$ if the elements of the separating system are paths (trees) in the graph $G$. In this paper, we focus on the size of the smallest vertex-separating path (tree) system for different types of graphs, including trees, grids, and maximal outerplanar graphs.

On Separating Path and Tree Systems in Graphs

TL;DR

This work studies vertex-separating path and tree systems in graphs, formalizing the parameters and and establishing tight bounds across key graph classes. The authors develop constructive labeling and decomposition techniques, deriving sharp results for complete bipartite graphs, grids, and maximal outerplanar graphs, and they prove a fundamental lower bound for trees with matching constructions. They further advance the theory by linking vertex-separating path systems to VC-dimension concepts to obtain polynomial lower bounds for certain minor-free graph classes, and by providing radius-based bounds for separating tree systems. The results deepen understanding of how efficiently vertex separation can be achieved with paths or trees and open several algorithmic and structural questions for broader graph families, including triangulations and graph products.

Abstract

We explore the concept of separating systems of vertex sets of graphs. A separating system of a set is a collection of subsets of such that for any pair of distinct elements in , there exists a set in the separating system that contains exactly one of the two elements. A separating system of the vertex set of a graph is called a vertex-separating path (tree) system of if the elements of the separating system are paths (trees) in the graph . In this paper, we focus on the size of the smallest vertex-separating path (tree) system for different types of graphs, including trees, grids, and maximal outerplanar graphs.
Paper Structure (9 sections, 18 theorems, 8 equations, 3 figures, 1 table)

This paper contains 9 sections, 18 theorems, 8 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Complete Bipartite Graphs
  4. Trees
  5. Grid Graphs
  6. Maximal Outerplanar Graphs
  7. Graph Classes With Polynomial Lower Bound
  8. Separating Tree Systems
  9. Open Problems

Key Result

Theorem 1

Let $G$ be a graph and let $X$ be a non-empty subset of $V(G)$. Then $X$ has a separating path system of size $\lceil \log{|X|} \rceil$ if and only if $X$ has a nice labeling.

Figures (3)

  • Figure 1: The tree $T_5$ obtained from the complete binary tree of height $4$.
  • Figure 2: Path $\Pi_3$ in blue colour and path $\Pi_1$ in orange colour in a $7\times16$ grid
  • Figure 3: An outerplanar graph in which the inner dual is a path. The black vertices represent the apex vertices of maximal fan subgraphs.

Theorems & Definitions (35)

  • Theorem 1: renyi1961random
  • Proposition 1
  • proof
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • Proposition 2
  • proof
  • Claim
  • proof
  • ...and 25 more