Twisted duality for embedded graphs
By Joanna A. Ellis-Monaghan, Iain Moffatt. 2009
64 citations recorded by the source corpus.
We consider two operations on an edge of an embedded graph (or equivalently a ribbon graph): giving a half-twist to the edge and taking the partial dual with respect to the edge. These two operations give rise to an action of S3 jE(G)j , the ribbon group, on G. The action of the ribbon group on embedded graphs extends the concepts of duality, partial duality and Petrie duality. We show that this ribbon group action gives a complete characterization of duality in that if G is any cellularly embedded graph with medial graph Gm, then the orbit of G under the group action is precisely the set of all graphs with medial graphs isomorphic (as abstract graphs) to Gm. We provide characterizations of special sets of twisted duals, such as the partial duals, of embedded graphs in terms of medial graphs and we show how dierent kinds of graph isomorphism give rise to these various notions of duality. The ribbon group action then leads to a deeper understanding of the properties of, and relationships among, various graph polynomials via the generalized transition polynomial which interacts naturally with the ribbon group action. ribbon group action on the set of embedded graphs by taking duals and twisting individual edges. The ribbon group action completes the classical concept of duality for embedded graphs, and we exploit this group action to unravel and explore connections between embedded graphs and their medial graphs, and to determine implications of these new relations. The ribbon group action allows us to: classify the twisted duals of an embedded graph; characterize all graphs with isomorphic medial graphs under various kinds of isomorphism; extend the concept of a Tait graph to all 4-regular graphs; and to determine new properties of polynomials of embedded graphs. There are many investigations into planar and abstract graphs that have not yet been fully extended to the context of graphs embedded in surfaces. Consider for example the classical relations among the medial graphs and the duals of plane graphs. Suppose that G is a plane graph, G is its dual, and Gm is its medial graph. The medial graph of G is exactly the medial graph of G, i.e. (G )m = Gm, where \=" denotes equality as plane graphs. In fact, the connection between geometric duals and medial graphs is a little stronger than this. The two graphs G and G are the only plane graphs that have Gm as their plane medial graphs, that is,
Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/96cc0a22be8aa4c1f570182510625b1a42b0869f
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In Discrete Mathematics and Combinatorics
In Mathematical Physics
Citations in the corpus
- On the interplay between embedded graphs and delta‐matroids (2016, background): “It was shown in [16] that G acts on the set X = {(G,A) : G a ribbon graph, A ⊆ E(G)} by g(G,A) := (G, A) for g ∈ G.”