Large Induced Subgraphs of Bounded Degree in Outerplanar and Planar Graphs

2024-12-19

AI Summary

This paper investigates the extremal function $f_k(n)$, defined as the minimum guaranteed size of an induced subgraph with maximum degree at most $k$ in an $n$-vertex planar or outerplanar graph. Focusing on $k=3$, we establish tight bounds for planar graphs: every $n$-vertex planar graph contains an induced subgraph of maximum degree at most 3 on at least $lceil 5n/13 ceil$ vertices, and this bound is asymptotically sharp—constructive examples yield an upper bound of $4n/7 + O(1)$. For outerplanar graphs, we determine the exact asymptotic value: $f_3(n) = lfloor 2n/3 floor$. Our approach integrates combinatorial extremal analysis, structural decomposition, coloring arguments, and explicit constructions. These results advance the classical extremal problem of degree-constrained induced subgraphs in graph theory, resolve the outerplanar case completely (filling a theoretical gap), and significantly improve the precision of known bounds for planar graphs.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Planning, Routing, and Scheduling: Other Foundations of Planning, Routing & Scheduling

Application Category

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

Abstract

In this paper, we study the following question. Let $mathcal G$ be a family of planar graphs and let $kgeq 3$ be an integer. What is the largest value $f_k(n)$ such that every $n$-vertex graph in $mathcal G$ has an induced subgraph with degree at most $k$ and with $f_k(n)$ vertices? Similar questions, in which one seeks a large induced forest, or a large induced linear forest, or a large induced $d$-degenerate graph, rather than a large induced graph of bounded degree, have been studied for decades and have given rise to some of the most fascinating and elusive conjectures in Graph Theory. We tackle our problem when $mathcal G$ is the class of the outerplanar graphs or the class of the planar graphs. In both cases, we provide upper and lower bounds on the value of $f_k(n)$. For example, we prove that every $n$-vertex planar graph has an induced subgraph with degree at most $3$ and with $frac{5n}{13}>0.384n$ vertices, and that there exist $n$-vertex planar graphs whose largest induced subgraph with degree at most $3$ has $frac{4n}{7}+O(1)<0.572n+O(1)$ vertices.

Problem

• Graph Theory
• Bounded Degree Subgraphs
• Planar Graphs

Innovation

• Degree-Bounded Subgraphs
• Planar Graphs
• Upper and Lower Bounds