On 3-Connected Planar Graphs with Unique Orientable Circuit Double Covers

TL;DR

The paper addresses the Orientable Strong Embedding Conjecture for $3$-connected planar graphs by extending a known cubic-case result: a graph has a unique orientable circuit double cover precisely when it is the dual of an Apollonian network. It introduces two local graph modifications, complete augmentation and complete truncation, and proves they preserve planarity and $3$-connectivity, while establishing a duality framework via $(G^*)^a \cong (G^t)^*$. Using these tools, it proves that Apollonian duals are exactly the $3$-connected planar graphs with a single orientable circuit double cover, linking embeddings on orientable surfaces to Apollonian networks. The approach combines topological insights with combinatorial constructions and leverages dualities to achieve a complete classification, aided by computational checks on Apollonian structures.

Abstract

A circuit double cover of a bridgeless graph is a collection of even subgraphs such that every edge is contained in exactly two subgraphs of the given collection. Such a circuit double cover describes an embedding of the corresponding graph onto a surface. In this paper, we investigate the well-known Orientable Strong Embedding Conjecture. This conjecture proposes that every bridgeless graph has a circuit double cover describing an embedding on an orientable surface. In a recent paper, we have proved that a 3-connected cubic planar graph G has exactly one orientable circuit double cover if and only if G is the dual graph of an Apollonian network. In this paper, we extend this result by demonstrating that this characterisation applies to any 3-connected planar graph, regardless of whether it is cubic.

Figures (8)

• Figure 1: A planar embedding of $K_{2,2,2}$
• Figure 2: A 3-connected planar graph $G$ (a) and the graph that results from augmenting the inner face of length 6 of $G$ (b)
• Figure 3: The cubical graph $G$ (a) and its complete augmentation $G^a$
• Figure 4: A $3$-connected planar graph $G$ (a) and the graph resulting from applying a truncation to $G$ at the vertex of degree 6 (b)
• Figure 5: The complete graph $G\cong K_4$ (a) and its complete truncation $G^t$ where the edges of Type 1 are coloured red and the edges of Type 2 are coloured blue (b)

Theorems & Definitions (18)

• Theorem 1.1
• Definition 3.1
• Definition 3.2
• Lemma 3.3
• proof
• Definition 3.4
• Definition 3.5
• Remark 3.6
• Lemma 3.7
• proof