ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-11

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Two plane embeddings of one connected planar graph have nonisomorphic dual multigraphs

Example

The abstract graph below has two plane embeddings whose dual multigraphs are distinguished by their degree multisets. For a finite multigraph with the endpoint map of Multigraphs, loops and directed graphs as variants distinct from the default finite simple graph, count an ordinary incident edge once and a loop twice in the degree. An isomorphism of such endpoint-map structures is a pair of vertex and edge bijections preserving endpoints, and therefore preserves the degree multiset.

Facts & Assumptions

Given: Let $G$ have vertices $s, t, a, b, c, w$ . Its edges form three internally disjoint $s$ - $t$ paths $P_{1} = s t$ , $P_{2} = s a t$ , and $P_{3} = s b c t$ , together with the bridge $s w$ .

[L1]

Every connected plane graph has a plane dual (Every connected plane graph has a plane dual multigraph, and when that dual is simple the reciprocal embedding identifies the double dual with the original graph).

[F1]

An endpoint-preserving multigraph isomorphism preserves the degree multiset because it bijects the incident edge-ends at every vertex, with a loop contributing two ends.

Verification

technique · constructive

1.1

Embed the three $s$ - $t$ paths in cyclic order. Write $f_{12}, f_{23}, f_{31}$ for the faces between the indicated path pairs. In the dual of The plane dual multigraph, with a vertex for each face and one crossing edge for each primal edge, the path lengths $1, 2, 3$ give respectively that many parallel dual edges between the two faces on each side of the path. Before accounting for $s w$ , the degrees of $f_{12}, f_{23}, f_{31}$ are $3, 5, 4$ . The bridge may be drawn into any one of these face sectors at $s$ ; it then gives a dual loop at that face.

L1construct

2.1

In one embedding draw $s w$ into $f_{12}$ ; the dual degree multiset is then ${5, 5, 4}$ . In another draw it into $f_{31}$ ; the dual degree multiset is ${3, 5, 6}$ . These multisets differ, since a loop contributes two to its incident degree. By [F1] the two dual multigraphs are nonisomorphic, although both arise from the same connected planar graph $G$ .

L1F1step 1.1discharge-construct∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, plane duals (standard reference, not scraped)
Jeff Erickson, Planar Graphs (standard reference, not scraped)