DEFINITION
If G=( ( ), ( )υ G ε G ) is a ribbon graph then we can construct a graph G =( ( ), ( )υ G ε G ) form G by replacing each edge of G with a line, and then contracting the vertices of G into points, such a graph G is called the core of G.
Notice that there is a natural correspondence between the edges of a ribbon graph and its core, and the vertices of a ribbon graph and its core.
DEFINITION
We say that two graphs are partial duals if they are cores of partially dual ribbon graphs.
Let G be a ribbon graph and A⊆ε (G). By the notation G we mean that A G is the A core of G where G is the core of G and A is the edge set of G that corresponds with A.A
We have seen that partially dual ribbon graphs can be characterized by the existence of an appropriate partially dual embedding. A corresponding result holds partial dual graphs.
To describe the corresponding result, we make the following definition.
DEFINITION
A partial dual embedding of graphs is a set{ , , ,G H% %Σ Ε}
Where Σ is a surface without boundary ,G H% %⊂ Σ are embedded graphs and E is a set of colored edges that are embedded in Σ such that
(1) Only the ends of each embedded edge in E meet G H% %∪ ⊂ Σ; (2) { , ,G H% % } is dual embedding;Σ
(3) Each edge in E is incident to one vertex in ( )υ G% and one vertex in ( )υ H%;
(4) There are exactly two edges of each color in E.
THEOREM 1
Two graphs G and 1 G are partial duals if and only if there exists a partial dual 2
embedding {G G% %1, 2, ,Σ Ε} such that for each i, G is obtained from i G% by adding an edge i
between the vertices of G%, that are incident with the two edges in E that have the same color, i for each color.
EXAMPLE 2
An example of partial dual embedding
Figure: 1
Where Σis the disjoint union of two spheres, G%1=({ , },{1})α β%% and G%2 =({ , , },{1})a b c% %% . Following the recipe in the theorem we recover the graphs.
G G{1}
Figure: 2
These graphs are indeed partial duals as they are cores of the following graphs respectively.
Figure: 3
We will now prove theorem:1. The idea behind the proof is construct a correspondence between partial dual embeddings of ribbon graphs and their (embedded) cores. It then follows by a theorem that the graphs constructed by this theorem are the cores of partially dual ribbon graphs.
PROOF
First suppose that G and 1 G are partial duals, so 2 G and 1 G are the cores of partially 2
dual ribbon graphs. Then by theorem: , there exists a partial dual embedding { ,G G% %1 2, ,Σ M}
such that Σ\ (Gυ %) M is an arrow-marked ribbon graph describing 2 ∪ G1 = Σ\ (υ G%) M is 1 ∪
an arrow-marked ribbon graph describing G : 2 G is the core of 1 G ; and 1 G is the core of 2 G .2
A partial dual embedding of graphs {G G% % ,E} can be constructed from {1, 2,Σ
1, 2, ,
G G% % M } in the following way: Let Σ G% be the canonically embedded core of 1 G% and 1 G% 2
let be the canonically embedded core of G%. Each arrow on 2 Σ meets exactly two vertices of
1 2
G% %. For each arrow, add an embedded edge between the two corresponding vertices of ∪G
the graph G% %1∪G2 ⊂ Σ which passes through this arrow. Color the edge with the color of the arrow that it passes through. The set of edges added in this way forms E.
We need to show that [ ,G G% %1 2, , }Σ Ε is indeed a partial dual embedding of graphs and
the graphs G and 1 G can be recovered from the partial dual embedding in the way described 2
by the theorem.
one vertex in V G% . The coloring requirement follows since there are exactly two edges of ( )2 each color in M and the edge colorings of E are induced from M .
Finally,G can be recovered from i G% M by adding edges between the marking i∪
arrows of the same color. Therefore, if u and v are vertices of G% which are marked with an i
arrow of the same color and u and v are vertices of G% which are marked with an arrow of the i
same color and u and v are the corresponding vertices of G%, then to construct the core of i G i
we need to add an edge between u and v. But since u and v are each incident with the edges in
E of the same color we need to add an edge between the vertices of G% that are incident with i the two edges in E of the same color. This is exactly the construction described in the statement of the theorem. Using this for each color gives G , completing the proof of i
necessity.
Conversely, suppose that {G G% % ,E} is a partial dual embedding and that 1, 2,Σ G and 1
G are obtained as described in the statement of the theorem. Construct a partial dual 2
embedding {G G% % ,1, 2,Σ M } of ribbon graph in the following way: take a small neighbourhood
in Σ of the embedded graph G% to form 1 G%; let 1 G =(2 Σ\G%1, ( )ε G% ); wherever an edge in E 1 meets a boundary of vertices add an arrow pointing in an arbitrary direction which is colored by the color of the edge in E. M is the set of such colored arrows.
To see that {G G% %1, , ,M } is a partial dual embedding, note that {2 Σ G G% %1, , } is a dual 2 Σ
embedding since {G G% %1, , } is, and that there exactly two arrows of each color since there 2 Σ are exactly two edges of each color in E.
Let G denote the ribbon graph described by the arrow-marked ribbon graph i G% M . i∪
Then G is the core of i G (since whenever an edge is added between two vertices of i G% in the i
formation of G , an edge is added between the corresponding vertices of i G% in the formation i
of G ). Finally, i G and 1 G are partial dual graphs since, by Theorem: 2 G and 1 G are partial 2
dual ribbon graphs.
The corollary below follows from the construction of a partial dual embedding in the proof above.
COROLLARY 3
If G and G are partial duals then the corresponding partial dual embedding as A constructed by Theorem: , is { \G A Gc, A\ ( ), , }φ Ac Σ E , where Ac =ε( ) \G A. Moreover, G (respectively G ) is obtained from \A G A (respectively c GA\ ( )φ Ac ) by adding an edge
between the vertices of \G A (respectively c GA \ ( )φ Ac ) that are incident with the two edges in E that have the same color for each color.