Excluding a large theta graph

TL;DR

This paper characterizes the structure of graphs that exclude large theta graphs as topological minors, especially focusing on $ heta_{t,t,t}$-free graphs, revealing their composition from planar graphs and certain sums.

Contribution

It provides a complete structural characterization of large $ heta_{a,b,c}$-free graphs, extending known results to new classes of forbidden subgraphs.

Findings

3-connected $ heta_{t,t,t}$-free graphs are formed by 3-summing graphs without long paths to planar graphs.

2-connected $ heta_{t,t,t}$-free graphs are constructed via 2- and 3-sums from these components.

The results generalize a theorem on graphs with bonds containing three specific edges.

Abstract

A theta graph, denoted $\theta_{a,b,c}$, is a graph of order $a+b+c-1$ consisting of a pair of vertices and three independent paths between them of lengths $a$, $b$, and $c$. We provide a complete characterization of graphs that do not contain a large $\theta_{a,b,c}$ as a topological minor. More specifically, we describe the structure of $\theta_{1,2,t}$-, $\theta_{2,2,t}$-, $\theta_{1,t,t}$-, $\theta_{2,t,t}$-, and $\theta_{t,t,t}$-free graphs where $t$ is large. The main result is a characterization of $\theta_{t,t,t}$-free graphs for large $t$. The $3$-connected $\theta_{t,t,t}$-free graphs are formed by $3$-summing graphs without a long path to certain planar graphs. The $2$-connected $\theta_{t,t,t}$-free graphs are then built up in a similar fashion by 2- and 3-sums. This result implies a well-known theorem of Robertson and Chakravarti on graphs that do not have a bond containing three specified edges.