Classification and Construction of Planar, 3-Connected Kronecker Products

Abstract

We give a complete classification of the Kronecker (i.e. direct) product graphs that are planar and $3$-connected (i.e. $3$-polytopal). They are all of the form \[H\wedge K_2,\] where $H$ is a $2$-connected graph, possibly non-planar, and satisfying specific properties that we will describe. Our proof is constructive, in the sense that we prescribe how to obtain all such graphs $H$, by adding a few edges in a specific way to a given planar, bipartite graph, that is either $3$-connected, or semi-hyper-$2$-connected. Moreover, for $H$ planar, we also give a more precise characterisation of this graph, regarding the number of its odd regions, and how they intersect. If $H\wedge K_2$ is a $3$-polytope, then we have $δ(H\wedge K_2)=3$, so that the connectivity of $H\wedge K_2$ is $3$, and the connectivity of $H$ is either $2$ or $3$. We also briefly discuss which Cartesian and strong products are $3$-polytopal.

Figures (21)

• Figure 1: Theorem \ref{['thm:2']}.
• Figure 2: Examples of $3$-polytopes $H$ as in Theorem \ref{['thm:1']}, and the corresponding $\mathcal{P}=H\wedge K_2$.
• Figure 3: The non-planar graph $H_n$, $n\geq 2$.
• Figure 4: Theorem \ref{['thm:3']}.
• Figure 5: Propositions \ref{['prop:cart']} and \ref{['prop:strong']}.

Theorems & Definitions (41)

• Proposition 1.1
• Proposition 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 1.5
• Theorem 1.6
• Theorem 1.7
• Proposition 1.9
• Proposition 1.10
• proof : Proof of Proposition \ref{['prop:2']}