Minor-Obstructions for Apex Sub-unicyclic Graphs

TL;DR

This paper characterizes and enumerates the minor-obstructions for classes of graphs that become sub-unicyclic after removing up to k vertices, providing structural insights and growth bounds.

Contribution

It identifies all minor-obstructions for 1-apex sub-unicyclic graphs and characterizes those for larger k, including enumeration and structural description.

Findings

Identified 29 minor-obstructions for 1-apex sub-unicyclic graphs.

Provided structural characterization of obstructions for larger k.

Established exponential lower bounds on the number of obstructions for each k.

Abstract

A graph is sub-unicyclic if it contains at most one cycle. We also say that a graph $G$ is $k$-apex sub-unicyclic if it can become sub-unicyclic by removing $k$ of its vertices. We identify 29 graphs that are the minor-obstructions of the class of $1$-apex sub-unicyclic graphs, i.e., the set of all minor minimal graphs that do not belong in this class. For bigger values of $k$, we give an exact structural characterization of all the cactus graphs that are minor-obstructions of $k$-apex sub-unicyclic graphs and we enumerate them. This implies that, for every $k$, the class of $k$-apex sub-unicyclic graphs has at least $0.34\cdot k^{-2.5}(6.278)^{k}$ minor-obstructions.