Combinatorial $n$-od covers of graphs

TL;DR

The paper introduces combinatorial $n$-od covers as a new tool to analyze plane continua, generalizes classic examples, and constructs indecomposable plane continua with specific od-like properties and arc subcontinua.

Contribution

It develops the concept of combinatorial $n$-od covers and constructs indecomposable plane continua with precise od-like characteristics, extending previous examples.

Findings

Introduces combinatorial $n$-od covers as a new analytical tool.

Constructs indecomposable plane continua that are simple $(n+1)$-od-like but not simple $n$-od-like.

Shows each non-degenerate proper subcontinuum is an arc.

Abstract

We introduce the notion of a combinatorial $n$-od cover, for $n \geq 3$, which is a tool that may be used to show that certain continua embedded in the plane are not simple $n$-od-like. Using this tool, we generalize a classic example of Ingram, and give a construction, for each $n \geq 3$, of an indecomposable plane continuum which is simple $(n+1)$-od-like but not simple $n$-od-like, and such that each non-degenerate proper subcontinuum is an arc. These examples may be compared with related constructions of Kennaugh [10].