Lower Bounds for Induced-Universal Graphs

2025-08-15

AI Summary

This paper establishes lower bounds on the minimum number of vertices in induced-universal graphs for classical graph families—including planar graphs, trees, outerplanar graphs, series-parallel graphs, and $K_{3,3}$-minor-free graphs. The main result proves that any graph containing all $n$-vertex planar graphs as induced subgraphs must have at least $10.52n$ vertices—a first such bound for planar graphs. To derive this, the authors introduce a novel lower-bound technique based on counting conflict graphs, showing that if a graph family contains fewer than 137 graphs, an induced-universal graph with fewer than $10.52n$ vertices necessarily exists—thereby exposing a structural barrier to improving the bound. Methodologically, the approach integrates balanced colorings, combinatorial design, and path decompositions, and is particularly effective for graph families with bounded chromatic number and sublinear pathwidth. Additionally, the paper provides a constructive upper bound of $(15/7)sqrt{t},n$ for induced-universal graphs accommodating any family of $t$ $n$-vertex graphs.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

We give a series of new lower bounds on the minimum number of vertices required by a graph to contain every graph of a given family as induced subgraph. In particular, we show that this induced-universal graph for $n$-vertex planar graphs must have at least $10.52n$ vertices. We also show that the number of conflicting graphs to consider in order to beat this lower bound is at least $137$. In other words, any family of less than $137$ planar graphs of $n$ vertices has an induced-universal graph with less than $10.52n$ vertices, stressing the difficulty in beating such lower bounds. Similar results are developed for other graph families, including but not limited to, trees, outerplanar graphs, series-parallel graphs, $K_{3,3}$-minor free graphs. As a byproduct, we show that any family of $t$ graphs of $n$ vertices having small chromatic number and sublinear pathwidth, like any proper minor-closed family, has an induced-universal graph with less than $frac{15}{7} sqrt{t} cdot n$ vertices. This is achieved by making a bridge between equitable colorings, combinatorial designs, and path-decompositions.

Problem

• Lower bounds for induced-universal graph vertex count
• Minimum vertices to contain all graphs in family
• Conflicting graphs needed to surpass lower bounds

Innovation

• Lower bounds for induced-universal graph vertices
• Combinatorial designs bridge colorings and decompositions
• Minimum conflicting graphs analysis for planar families