The minimal nonplanar strong digraphs

TL;DR

The paper addresses the problem of identifying minimal strong nonplanar digraphs by reducing to orientations of almost-planar undirected graphs. It provides a structural taxonomy: seven core almost-planar types (Mobius_chain, double_wheel, conch, scallop, mussel, clam, whelk) plus notable eight-vertex graphs $U_8$, $V_8$, and $W_8$, and analyzes how these yield (or fail to yield) Kuratowski digraphs under directed subdivisions. Outerplanar and series-parallel obstructions are characterized via directed subdivisions of specific subdigraphs such as $K_4$ and $2k$-wheels, and a detailed almost-planar graph classification underlies the orientation problem. A combination of combinatorial, parity, and 2-SAT techniques is developed to decide when an almost-planar graph admits a good orientation, highlighting both tractable cases and the inherent complexity in Möbius chains. The results collectively advance understanding of the landscape of minimal strong nonplanar digraphs and the conditions under which they can be realized as oriented almost-planar graphs.

Abstract

Kuratowski's theorem says that the minimal (under subgraph containment) graphs that are not planar are the subdivisions of $K_5$ and of $K_{3,3}$. Here we study the minimal (under subdigraph containment) strongly-connected digraphs that are not planar. We also find the minimal strongly-connected non-outerplanar digraphs and the minimal strongly-connected non-series-parallel digraphs.

Figures (7)

• Figure 1: A Möbius chain and a double wheel.
• Figure 2: A conch, a mussel, a scallop, a clam and a whelk.
• Figure 3: Obstructions for outerplanar.
• Figure 4: A five-rung Möbius ladder.
• Figure 5: A Möbius chain.