A characterization of partially dual graphs

Abstract

In this paper, we extend the recently introduced concept of partially dual ribbon graphs to graphs. We then go on to characterize partial duality of graphs in terms of bijections between edge sets of corresponding graphs. This result generalizes a well known result of J. Edmonds in which natural duality of graphs is characterized in terms of edge correspondence, and gives a combinatorial characterization of partial duality.Comment: V2: the statement of the main result has been changed. To appear in JGT

On the Seifert graphs of a link diagram and its parallels

“…We will use the construction of a partial dual from [19]. The idea behind this construction is that given a set A of edges the partial dual with respect to A can be formed by 'hiding' the edges not in A and replacing them with marking arrows, forming the dual of the resulting arrow marked ribbon graph, and then revealing the hidden edges.…”

Bipartite partial duals and circuits in medial graphs

“…Roughly speaking, a partial dual is obtained by forming the geometric dual with respect to only a subset of edges of an embedded graph (a formal definition is given Subsection 2.3). Partial duality appears to be a fundamental operation on embedded graphs and, although it has only recently been introduced, it has found a number of applications in graph theory, topology, and physics (see, for example, [3,5,6,8,9,10,11,12,14,15]). While geometric duality always preserves the surface in which a graph is embedded, this is not the case for the more general partial duality.…”

Eulerian partial duals of plane graphs

“…Partial duality was further generalized to twisted duality by Ellis-Monaghan and Moffatt in [5]. Both are far-reaching extensions of geometric duality and have found a number of significant applications in graph theory, topology, and physics (see, e.g., [7][8][9][10][11][12][13][14]).…”