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arXiv · 1902.02231

Minor-Obstructions for Apex Sub-unicyclic Graphs

Alexandros Leivaditis·Alexandros Singh·Giannos Stamoulis·Dimitrios Thilikos·Konstantinos Tsatsanis·Vasiliki Velona
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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.

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Alexandros Leivaditis, Alexandros Singh, Giannos Stamoulis, Dimitrios Thilikos, Konstantinos Tsatsanis, Vasiliki Velona. 2019-02-06. Minor-Obstructions for Apex Sub-unicyclic Graphs. https://arxiv.org/abs/1902.02231

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