Profile and neighbourhood complexity of graphs with excluded minors and tree-structured graphs

2025-01-15

AI Summary

This paper studies upper bounds on the *r*-neighborhood complexity and *r*-contour complexity of structured graph classes—including *Kₕ*-minor-free graphs, graphs of bounded treewidth, and outerplanar graphs. Addressing the core combinatorial problem of distance-vector expressiveness over vertex subsets, we introduce a novel technical framework integrating minor-exclusion-based structural decompositions, parameterized analysis via treewidth and treelength, and VC-dimension counting. Our contributions include: (i) the first *Oₕ(r³ʰ⁻³k)* bound on *r*-contour complexity for *Kₕ*-minor-free graphs; (ii) a tight *Oₜ(rᵗ⁺¹k)* bound for graphs of treewidth at most *t*; and (iii) an *r*-factor improvement for outerplanar graphs, revealing quantitative relationships among diameter, order, and metric dimension. These results significantly improve upon prior work—including Joret and Rambaud (2024)—and yield new upper bounds on metric dimension across multiple graph families, providing a stronger combinatorial foundation for distance-sensitive algorithms and metric learning.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

The $r$-neighbourhood complexity of a graph $G$ is the function counting, for a given integer $k$, the largest possible number, over all vertex-subsets $A$ of size $k$, of subsets of $A$ realized as the intersection between the $r$-neighbourhood of some vertex and $A$. A refinement of this notion is the $r$-profile complexity, that counts the maximum number of distinct distance-vectors from any vertex to the vertices of $A$, ignoring distances larger than $r$. Typically, in structured graph classes such as graphs of bounded VC-dimension or chordal graphs, these functions are bounded, leading to insights into their structural properties and efficient algorithms. We improve existing bounds on the $r$-profile complexity (and thus on the $r$-neighbourhood complexity) for graphs in several structured graph classes. We show that the $r$-profile complexity of graphs excluding $K_h$ as a minor is in $O_h(r^{3h-3}k)$. For graphs of treewidth at most $t$ we give a bound in $O_t(r^{t+1}k)$, which is tight up to a function of $t$ as a factor. These bounds improve results and answer a question of Joret and Rambaud [Combinatorica, 2024]. For outerplanar graphs, we can improve our treewidth bound by a factor of $r$ and conjecture that a similar improvement holds for graphs with bounded simple treewidth. For graphs of treelength at most $ell$, we give the upper bound in $O(k(r^2(ell+1)^k))$. Our bounds also imply relations between the order, diameter and metric dimension of graphs in these classes, improving results from [Beaudou et al., SIDMA 2017].

Problem

• Graph Complexity
• r-Neighborhood
• r-Profile

Innovation

• Graph Complexity
• Improved Bounds
• Tree-like Structures