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[Added another complementary question below.]

Motivation

The 1-skeleton of the triangular bipyramid seems to be the smallest connected planar graph $G$ with the following

Property: There is a cycle $\gamma$ ($\color{red}{\mathsf{red}}$) in $G$ with exactly two connected components $G_1, G_2$ of $G - \gamma$ such that for every graph embedding $\pi$ of $G$ into the plane $\mathbb{R}^2$ the component $G_1$ ($\color{blue}{\mathsf{blue}}$) is contained in the interior of $\pi(\gamma)$ and the other component $G_2$ ($\color{black}{\mathsf{black}}$) is contained in the exterior of $\pi(\gamma)$ – or vice versa.

enter image description here

Definition: A cycle $\gamma$ in the graph $G$ is a Jordan cycle if $G - \gamma$ splits up into exactly two connected components $G_1, G_2$ such that for every graph embedding $\pi$ of $G$ into the plane $\mathbb{R}^2$ the component $G_1$ is contained in the interior of $\pi(\gamma)$ and the other component $G_2$ is contained in the exterior of $\pi(\gamma)$ – or vice versa.

Questions

  1. (How) can the property of being a Jordan cycle $\gamma$ be defined purely combinatorial, without mentioning graph embeddings $\pi$ and Jordan curves $\pi(\gamma)$?

  2. (How) can planar graphs containing a Jordan cycle be characterized purely combinatorially?

  3. ADDED: Can planar graphs be characterized in which every cycle is a Jordan cycle?
show/hide this revision's text 2 edited body; added 3 characters in body

Motivation

The 1-skeleton of the triangular bipyramid seems to be the smallest connected planar graph $G$ with the following

Property: There is a cycle $\gamma$ ($\color{red}{\mathsf{red}}$) in $G$ with exactly two connected components $G_1, G_2$ of $G - \gamma$ such that for every graph embedding $\pi$ of $G$ into the plane $\mathbb{R}^2$ the component $G_1$ ($\color{grey}{\mathsf{grey}}$) $\color{blue}{\mathsf{blue}}$) is contained in the interior of $\pi(\gamma)$ and the other component $G_2$ ($\color{black}{\mathsf{black}}$) is contained in the exterior of $\pi(\gamma)$ – or vice versa.

enter image description here

Definition: A cycle $\gamma$ in the graph $G$ is a Jordan cycle if there are $G - \gamma$ splits up into exactly two connected components $G_1, G_2$ of $G - \gamma$ such that for every graph embedding $\pi$ of $G$ into the plane $\mathbb{R}^2$ the component $G_1$ is contained in the interior of $\pi(\gamma)$ and the other component $G_2$ is contained in the exterior of $\pi(\gamma)$ – or vice versa.

Questions

  1. (How) can the property of being a Jordan cycle $\gamma$ be defined purely combinatorial, without mentioning graph embeddings $\pi$ and Jordan curves $\pi(\gamma)$?

  2. (How) can planar graphs containing a Jordan cycle be characterized purely combinatorially?
show/hide this revision's text 1

Jordan-like cycles in graphs

Motivation

The 1-skeleton of the triangular bipyramid seems to be the smallest connected planar graph $G$ with the following

Property: There is a cycle $\gamma$ ($\color{red}{\mathsf{red}}$) in $G$ with exactly two connected components $G_1, G_2$ of $G - \gamma$ such that for every graph embedding $\pi$ of $G$ into the plane $\mathbb{R}^2$ the component $G_1$ ($\color{grey}{\mathsf{grey}}$) is contained in the interior of $\pi(\gamma)$ and the other component $G_2$ ($\color{black}{\mathsf{black}}$) is contained in the exterior of $\pi(\gamma)$ – or vice versa.

enter image description here

Definition: A cycle $\gamma$ in the graph $G$ is a Jordan cycle if there are exactly two connected components $G_1, G_2$ of $G - \gamma$ such that for every graph embedding $\pi$ of $G$ into the plane $\mathbb{R}^2$ the component $G_1$ is contained in the interior of $\pi(\gamma)$ and the other component $G_2$ is contained in the exterior of $\pi(\gamma)$ – or vice versa.

Questions

  1. (How) can the property of being a Jordan cycle $\gamma$ be defined purely combinatorial, without mentioning graph embeddings $\pi$ and Jordan curves $\pi(\gamma)$?

  2. (How) can planar graphs containing a Jordan cycle be characterized purely combinatorially?