Self-Dual Embeddings of Composition Graphs (original) (raw)

DUAL GRAPHS

isara solutions, 2017

A plane graph g partitions the rest of the plane into a number of connected regions, the closures of these regions are called the face of G, shows a plane graph with six faces f1, f2, f3, f4, f5 and f6. The notion of a face applies also to embeddings of graphs on other surfaces. We shall denote by F(G) and  (G), respectively, the set of faces and the number of faces of a plane graph G.

Obstructions for two-vertex alternating embeddings of graphs in surfaces

European Journal of Combinatorics, 2017

A class of graphs that lies strictly between the classes of graphs of genus (at most) k − 1 and k is studied. For a fixed orientable surface S k of genus k, let A k xy be the minor-closed class of graphs with terminals x and y that either embed into S k−1 or admit an embedding Π into S k such that there is a Π-face where x and y appear twice in the alternating order. In this paper, the obstructions for the classes A k xy are studied. In particular, the complete list of obstructions for A 1 xy is presented.

Embedding Half-Edge Graphs in Punctured Surfaces

arXiv: Combinatorics, 2017

It is known that graphs cellularly embedded into surfaces are equivalent to ribbon graphs. In this work, we generalize this statement to broader classes of graphs and surfaces. Half-edge graphs extend abstract graphs and are useful in quantum field theory in physics. On the other hand, ribbon graphs with half-edges generalize ribbon graphs and appear in a different type of field theory emanating from matrix models. We then give a sense of embeddings of half-edge graphs in punctured surfaces and determine (minimal/maximal) conditions for an equivalence between these embeddings and half-edge ribbon graphs. Given some assumptions on the embedding, the geometric dual of a cellularly embedded half-edge graph is also identified.

On the surface duality of linear graphs

Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics, 1965

iFe br ua ry 3, 1964) TH EOR EM : A 1·1 co rres po nd e nce be t wee n the ed ge s of two co nn ec ted g ra ph s is a uu a lit y with re s pe c t to s u me po lyh edral s urface e mb e ddin g if and o nl y if for ea c h ve rte x v uf e ac h gra ph , th e ed ges whi ch mee t v co rres po nd in th e oth e r graph to th e ed ge s of a s ub graph G,· whi c h is co nn ec ted a nd whi c h has a n e ve n numb e r of it s e d ge·e nd s to e ac h of it s ve rti ces (w he re if a n ed ge mee ts va t bo th e nd s it s ima ge in G,. is co unt ed twi c e) . Us in g th e Eul e r furmul a, th e charac te ri s ti c of the s urfa ce is de termin ed by th e two gra ph s. Thu s, th e th eore m ge ne rali zes a var ia ti o n of th e H. Whitn ey conditi o n fur a gra ph to be pl a nar.

Partial duality and closed 2-cell embeddings

In 2009 Chmutov introduced the idea of partial duality for embeddings of graphs in surfaces. We discuss some alternative descriptions of partial duality, which demonstrate the symmetry between vertices and faces. One is in terms of band decompositions, and the other is in terms of the gem (graph-encoded map) representation of an embedding. We then use these to investigate when a partial dual is a closed 2-cell embedding, in which every face is bounded by a cycle in the graph. We obtain a necessary and sufficient condition for a partial dual to be closed 2-cell, and also a sufficient condition for no partial dual to be closed 2-cell.

Imbeddings of the tensor product of graphs where the second factor is a complete graph

Discrete Mathematics, 1998

In this paper we present genus result for the tensor product of graphs where the second factor is the complete graph Km and m is a power of 2. We are going to view the second factor as a Cayley graph and we will use an imbedding technique that combines surgery and voltage graph theory. In this imbedding technique we start with a suitable imbedding of H on some surface and then modify the edges of H according to the nature of K., to get a voltage graph H*. The covering graph of this voltage graph H* will be the desired tensor product of H with K,,. The genus of H ® K.,, where H is a graph with a quadrilateral imbedding and also H is either, bipartite, with bichromatic dual, or with straight-ahead imbedding is given

An obstruction to embedding graphs in surfaces

Discrete Mathematics, 1989

It is shown that the genus of an embedding of a graph can be determined by the rank of a certain matrix. Several applications to problems involving the genus of graphs are presented.