Self-dual Embeddings of K_{4m,4n} in Different Orientable and Nonorientable Pseudosurfaces with the Same Euler Characteristic

A proper embedding of a graph G in a pseudosurface P is an embedding in which the regions of the complement of G in P are homeomorphic to discs and a vertex of G appears at each pinchpoint in P; we say that a proper embedding of G in P is self dual if there exists an isomorphism from G to its dual graph. We give an explicit construction of a self-dual embedding of the complete bipartite graph K_{4m,4n} in an orientable pseudosurface for all $m, n\ge 1$; we show that this embedding maximizes the number of umbrellas of each vertex and has the property that for any vertex v of K_{4m,4n}, there are two faces of the constructed embedding that intersect all umbrellas of v. Leveraging these properties and applying a lemma of Bruhn and Diestel, we apply a surgery introduced here or a different known surgery of Edmonds to each of our constructed embeddings for which at least one of m or n is at least 2. The result of these surgeries is that there exist distinct orientable and nonorientable pseudosurfaces with the same Euler characteristic that feature a self-dual embedding of K_{4m,4n}.


Introduction
To us, a graph is a finite and connected multigraph, and a surface is a compact and connected 2manifold without boundary; we let G denote a graph and S denote a surface. A cellular embedding of G in S is an embedding in which the complement of G in S is a disjoint union of regions (called faces), each of which is homeomorphic to a disc. We will let G → S denote a cellular embedding of G in S. Following [11], given G → S, we define the dual graph G * and the dual embedding (G → S) * as follows: the vertices of G * are the "centers" of the faces of G in S, and each edge e * corresponds bijectively to an edge e of G and connects the vertice(s) of G * corresponding to the face(s) on either side of e. We say that two embeddings of G in surfaces S and T , denoted i : G → S and j : G → T , are equivalent if there is an orientation-preserving homeomorphism f : S → T such that f • i = j. Furthermore, G → S is self dual if the cellular structure of vertices, edges, and faces (commonly called 0-cells, 1-cells, and 2-cells, respectively) given by G in S is isomorphic to the cellular structure given by G * in S; this implies that G and G * are isomorphic. By [11, §1.4.8], this also implies that ((G → S) * ) * and G → S are equivalent embeddings. An example of a self-dual graph embedding in a surface is given in Figure 1. The purpose of this article is to begin an investigation of self-dual embeddings and embeddability of graphs in pseudosurfaces. To place this discussion in some context, we describe some of the research concerning self-dual embeddings of graphs in surfaces, which is quite rich. Recently, Abrams and Slilaty have merged the study of self-dual graph embeddings in surfaces with the study of symmetries of cellular decompositions of surfaces [1]. Moreover, the self-dual graph embeddings in various surfaces have been cataloged from different viewpoints: in the sphere by Abrams and Slilaty [1], and by B. Servatius and H. Sevatius [16,17]; in the projective plane by Abrams and Slilaty [1], and by Archdeacon and Negami [7]; and in all other surfaces of Euler characteristic at least −1 [1]. Archdeacon and Hartsfield in [6] gave results concerning the orientable and nonorientable self-dual embeddability of complete bipartite graphs in surfaces; the current work can be viewed as the first step in extending their results to pseudosurfaces. Of similar interest is another article by Archdeacon [3] in which he shows that one can use ordinary voltage graph theory to create self-dual graph embeddings in surfaces, and other non self-dual embeddings in surfaces for which the dual is predictable and has very specific properties.
Following [2] a closed, connected pseudosurface is a connected topological space obtained from a disjoint union of surfaces via a finite number of point identifications, called pinches. A surface is therefore a special case of a pseudosurface. The points that are identified with other points are called pinchpoints. A small-enough neighborhood of a pinchpoint is homeomorphic to the union of discs identified at a point; each identified disc is called an umbrella of the pinchpoint. A proper embedding of a graph G in a pseudosurface P is an embedding in which each of the regions of the complement of G in P is homeomorphic to a disc and a vertex of G appears at each pinchpoint of P . We shall let G → P denote a proper embedding of G in P . We let F (G → P ) denote the set of faces of G → P . Given G → P , the definitions of the dual graph and the dual embedding of G → P are immediate natural extensions of the definition of the dual graph and dual embedding of G → S, respectively: (G → P ) * captures the incidence of faces and edges of G in P . However, as evidenced by Figure 2, ((G → P ) * ) * is not necessarily well defined since (G → P ) * is not a proper embedding if P has any pinchpoints. We therefore give a weaker notion of graph self-duality for pseudosurfaces. For a pseudosurface P with at least one pinchpoint, we say that G → P is self dual if G * is isomorphic to G, which still requires that the incidence of faces and edges in G → P is isomorphic to the incidence of vertices and edges in G.
We should note that there has been some attention given to the study of embeddings and embeddability of graphs in pseudosurfaces. Archdeacon's survey article [4, §5.7] and the introduction of [2] (which itself is about the embeddability of graphs in pseudosurfaces) contain some relevant references. Among other relevant works is [5], in which Archdeacon and Bonnington give the list of 21 graphs that form all obstructions of embeddability of cubic graphs in the pinched sphere. Part of the study of graph embeddings in pseudosurfaces has been about proving theorems in design theory. Among these efforts are works by Garman [10] and White [18]. Finally, the current authors and E. Rarity in [14] found the smallest simple graphs with a self-dual embedding in a pseudosurface with at least one pinchpoint. In Section 2, we construct self-dual embeddings of K 4m,4n in orientable pseudosurfaces for all m, n ≥ 1. In Section 3, we introduce our surgery and state that of Edmonds, including the necessary background information. Finally, in Section 4, we apply the surgeries of Section 3 to our embeddings from Section 2 to produce self-dual embeddings of K 4m,4n in different orientable and nonorientable pseudosurfaces with the same Euler characteristic.
2. Orientable self-dual embeddings of K 4m,4n in pseudosurfaces 2.1. Background necessary for our embeddings of K 4m,4n Recall that G is a finite and connected multigraph. We orient the edges of G and let → E(G) denote the set of directed edges of G; each directed edge we will call a dart on an edge of G. For each edge e of G, we will call one of the two corresponding darts the positive dart and the other the negative dart; the positive dart on an edge is assumed to correspond to the given orientation on that edge. For e ∈ E(G), we will let e also denote the positive dart on that edge and e − the negative dart. If G is a simple graph, then for v 1 , v 2 ∈ V (G), we let {v 1 , v 2 } denote an undirected edge joining v 1 and v 2 , and we let v 1 v 2 denote the dart from v 1 to v 2 . If e ∈ → E(G) = v 1 v 2 , then we say that the head of e and tail of e, denoted h(e) and t(e), are v 2 and v 1 , respectively; e − = v 2 v 1 , and (e − ) − = e. A walk in a graph G is a sequence of darts e 1 e 2 . . . e n , which in general may include negative darts, such that h(e i ) = t(e i+1 ). If h(e n ) = t(e 1 ), then we say that W is a closed walk. We will use a sequence of vertices to denote a walk in a simple graph since a simple graph has at most one edge joining two vertices; if a walk W = e 1 e 2 . . . e n satisfies h(e i ) = v i and t(e i ) = v i−1 , then we may let W be denoted by v 0 v 1 . . . v n . If W 1 = e 1 . . . e n and W 2 = e ′ 1 . . . e ′ m are walks having the property that h(e n ) = t(e ′ 1 ), then we let W 1 W 2 = e 1 . . . e n e ′ 1 . . . e ′ m . A surface or pseudosurface is said to be orientable if it can be triangulated by a proper graph embedding whose faces are compatibly oriented.
One way to encode a proper graph embedding in an orientable surface is with a description of a cyclic permutation of the edges incident to each vertex, with a convention that all cyclic permutations capture the edges in a counterclockwise or clockwise order; the same order must apply to all vertices in order to combinatorially capture the embedding. This is called a rotation scheme or rotation system. In order to describe a proper embedding in a pseudosurface with pinchpoints this way, one must be sure to describe the cyclic ordering of edges incident to each vertex v in each umbrella of v. In the case of proper embeddings in orientable pseudosurfaces with pinchpoints, one must add additional information to the encoding; for our purposes, this additional information will be captured in the collection of facial boundary walks. In the case of simple graphs, since there are no parallel edges joining two vertices, we may characterize the cyclic permutations of edges incident to a vertex by listing its neighboring vertices instead. We give here an example of a rotation of a vertex v in a simple graph with the neighboring vertices of v representing the edges appearing in each of three umbrellas represented by distinct parenthetical encapsulations.
v : For a complete treatment of rotation schemes for graphs embedded in surfaces, including nonorientable surfaces, the reader is referred to [11]. Since we will not be representing nonorientable embeddings as rotation schemes, we do not introduce the corresponding notation.
Let A = {a 0 , ..., a r−1 } and B = {b 0 , ..., b s−1 } be the bipartition sets of K r,s , where r ≤ s. We say that the dart a k b ℓ has slope ℓ − k (reduced modulo s).

The construction
Definition 1. Consider G → P . We say that a facial boundary walk F covers a vertex v if F visits every umbrella of v, and we say that G → P is doubly covered if every vertex of G is covered by two different facial boundary walks.
Consider G → P . If an umbrella of a vertex v contains only one or two neighbors of v, then the dual graph of that embedding will contain a loop or two parallel edges, respectively. Moreover, if an umbrella of v contains precisely three neighbors of v, then the dual graph contains a triangle. Thus, since K 4m,4n is a simple triangle-free graph, every umbrella of a vertex in a self-dual embedding of K 4m,4n must include at least four ends of edges incident to four distinct neighbors of that vertex, and it follows that the maximum number of umbrellas at any vertex of degree d is ⌊d/4⌋.
. We call an embedding K 4m,4n → P full if every vertex in A has n umbrellas and every vertex in B has m umbrellas.
It is immediate from Definition 2 and the preceding paragraph that if a proper embedding of K 4m,4n is full, then we may not add any more umbrellas to any more vertices and still expect to have a proper embedding of K 4m,4n featuring a simple dual graph without any cycles of length 3.
Theorem 2.1. There exists a doubly-covered, full, self-dual embedding of K 4m,4n in an orientable pseudosurface for all m, n ≥ 1.
We show that the collection forms a family of facial boundary walks that yields a self-dual embedding of K 4m,4n in an orientable pseudosurface with the desired properties.
Note that each pair of X-walks is edge-disjoint and each pair of Y -walks is edge-disjoint. The intersection of any X-walk with any Y -walk is given by the undirected edge below, where i and j satisfy 0 ≤ i ≤ m − 1 and 0 ≤ j ≤ n − 1, respectively.
From this it is clear that each undirected edge of the form {a k , b ℓ } is contained in precisely one X-walk and precisely one Y -walk, so when 2-cells are glued following the facial boundary walks, what results is a 2-complex, homeomorphic to a pseudosurface, whose 1-skeleton is K 4m,4n . Take each walk to be oriented from left to right as written above; then every dart a k b ℓ of even slope is oriented in the forward direction in an X-walk and in the reverse direction in a Y -walk. Similarly, every dart a k b ℓ of odd slope is oriented in the reverse direction in an X-walk and in the forward direction in a Y -walk. Thus, since the orientations are compatible, the embedding is in an orientable pseudosurface. In the dual graph of this embedding, the vertices corresponding to the faces bounded by X-walks form the independent set of size 4m in K 4m,4n and the vertices corresponding to the faces bounded by Y -walks form the independent set of size 4n. The intersections shown above verify that every vertex corresponding to an X-walk is adjacent to every vertex corresponding to a Y -walk. Thus, the embedding is self dual.
Next, we consider the rotation scheme for this embedding of K 4m,4n . Note that if an oriented facial walk contains the sequence (· · · u v w · · · ), then the rotation of v contains a permutation with the transition uw. The rotation at each vertex decomposes into a disjoint union of permutations of length four, and each permutation corresponds to an umbrella in the embedding. The complete rotation scheme is given below, where p and q satisfy 0 ≤ p ≤ 2m − 1 and 0 ≤ q ≤ n − 1, respectively; the first transition listed in each permutation corresponds to a sequence from an Xwalk.
For each vertex v, every umbrella of v contains 4 edges, each incident to a distinct neighbor of v. Thus, each vertex in A has m umbrellas and each vertex in B has n umbrellas, so the embedding is full.
Finally, note that a 4i and a 4i+1 are both covered by X i,0 and X i,1 , a 4i+2 and a 4i+3 are both covered by X i,2 and X i,3 , b 2j and b 4n−2j−1 are both covered by Y j,0 and Y j,1 , and b 2j+1 and b 4n−2j−2 are both covered by Y j,2 and Y j,3 , so the embedding is doubly covered. This completes the proof. Example 1. The following closed walks form the facial boundary walks of the orientable self-dual embedding of K 8,12 described in Theorem 2.1. 1 : (a 0 b 6 a 1 b 7 a 0 b 8 a 1 b 9 a 0 b 10 a 1 The rotation scheme for this embedding of K 8,12 is given below. Figure 3 shows a neighborhood of a 0 , and Figure 4 shows a neighborhood of b 1 . In Figures 3 and 4, we use the distinction of hollow and solid vertices to highlight the bipartition of the vertices of K 8,12 .

Background necessary for Edmonds' surgery and ours
The content of this subsection is adapted from [8,Section 2]. Recall that → E(G) denotes the set of darts (directed edges) of G, and let E(G) denote the real vector space of all functions φ : A cycle is a connected 2-regular graph. When W = e 1 e 2 . . . e n is a closed walk (which may include negative directed edges) traversing every edge it traverses at most once, we call the function mapping the positive directed edges of W to  1, the negative directed edges of W to −1, and all other directed edges of G to 0 an oriented circuit. The subspace of E(G) generated by the set of oriented circuits of G is called the oriented cycle space. Recall that a surface S is orientable if it can be triangulated by a graph embedding whose faces can be compatibly oriented. As pointed out in [8], equivalent conditions are that every triangulation has this property, and that the surface does not contain a Möbius strip. We say that a family of walks W double covers the edges of G if each edge is traversed twice in total by the walks of W.
For a walk W , we let  Following [1] we say that a pseudosurface P is face connected if any proper embedding of a connected graph in P induces a 2-complex K such that for any faces f a , f b of K, there is a sequence of faces f a = f 1 f 2 . . . f n = f b such that for each i ∈ {1, 2 . . . n − 1}, f i and f i+1 share a common boundary edge. Note that if G → P is an embedding in a non-face-connected pseudosurface, then the dual graph is not connected. For the remainder of this article, P shall denote a face-connected pseudosurface.
We let χ(P ) denote the Euler characteristic of P , which, as an invariant of P , depends on neither G nor any proper embedding of G in P . Given G → P ,

Edmonds' surgery
For a walk W , let W be the walk consisting of the darts opposite the darts of W , and appearing in reverse order relative to their counterparts in W . For example: if W = e − 1 e 2 e − 3 , then W = e 3 e − 2 e 1 . We give a somewhat revised proof of Theorem 3.1 for the sake of completeness since the original statement of [9, Theorem 1] did not include all the conclusions that one could derive from its proof. Proof. For each face of G → P , choose one of the two possible facial boundary walks, and let W be the set of these chosen facial boundary walks. Choose two umbrellas U 1 , U 2 of v intersected by f , and choose one intersection of f with each chosen umbrella. Let U * (v) denote the intersection of an arbitrarily small open neighborhood of v with U 1 ∪ U 2 , whose intersection with G consists only of v and the ends of edges incident to v. We let f i denote the chosen intersection of f with Let W denote the chosen facial boundary walk of f and let W = ω 1 ω 2 for closed walks ω 1 and ω 2 , as in Figure 5; ω 1 begins by traversing edge end 4 and ends after traversing edge end 1, and ω 2 begins by traversing edge end 2 and ends after traversing edge end 3. We surgically modify W to produce another facial boundary walk; let W ′ = ω 1 ω 2 (as in Figure 5), and let To see that the dual graphs of G → P and G → P ′ are identical, note that the darts of W and the darts of W ′ are darts on identical edges of G. Therefore, the incidence of faces and edges of G → P is the same as that of G → P ′ . It follows that the dual graphs of G → P and G → P ′ are identical and that P is also face connected.
To show that there is one fewer umbrella of v, we define an auxiliary graph that captures the incidence of certain regions of G → P with G. Let R 1 = U 1 \ f 1 and R 2 = U 2 \ f 2 . We define a graph H whose vertex set is {f 1 , f 2 , R 1 , R 2 } and with four edges e 1 , e 2 , e 3 , e 4 . Consider Figure 6; we define the edge e i to be incident to f j and R k if the edge end i is common to the boundaries of f j and R k . The graph H is comprised of two disjoint 2-cycles, one corresponding to each umbrella Figure 6. The subset U * (v) of P . The graph H is drawn in gray. Each edge e i is drawn transversely crossing the edge end i.
Because of the reversal of ω 2 to produce the facial walk W ′ , W ′ traverses the following ordered pairs of edge ends as W ′ enters and exits U * (v): 1 and then 3, and 2 and then 4. Let f ′ be the face of G → P ′ bounded by W ′ . We define f ′ 1 and f ′ 2 to be the regions of f ′ ∩ U * (v) bounded by edge ends 1 and 3 and 2 and 4, respectively. Note that the process of producing W ′ from W does not affect the boundaries of R 1 and R 2 . We define another analogous graph H ′ for G → P ′ whose vertex set is It is easy to see that this graph is a single cycle, and so we conclude that there is one less umbrella of v in G → P ′ .
Lastly, since |W| = |W ′ |, and G is unaffected by the surgery that created W ′ , we may conclude that χ(P ) = χ(P ′ ). Proof. This is an application of [8, Lemma 7]. The reader is advised that the surgery appearing in the proof of Theorem 3.1 is the same surgery that appears in the proof of [8, Lemma 7], which Bruhn and Diestel show results in a nonorientable pseudosurface.

Our surgery
Our surgery, which is given in the proof of Theorem 3.2, is applicable to fewer embeddings than the surgery of Edmonds described in Theorem 3.1 because it requires a face of G → P to intersect three umbrellas of a pinchpoint vertex v, whereas Edmonds' surgery requires two. However, our surgery does have the advantage that the orientability of the resulting pseudosurface is the same as the orientability of P . By Corollary 3.1, Edmonds' surgery necessarily produces a nonorientable embedding, no matter the orientability of the embedding to which Edmonds' surgery is applied.
Theorem 3.2. Consider G → P having a pinchpoint vertex v with the property that three umbrellas of v are intersected by the same face f . There exists a proper embedding of G in a different face-connected pseudosurface P ′ such that: the dual graphs of G → P and G → P ′ are identical, P ′ has the same orientability as P , there are two fewer umbrellas of v in G → P ′ , and χ(P ) = χ(P ′ ).
Proof. For each face of G → P , choose one of the two possible facial boundary walks, and let W be the set of chosen facial boundary walks. Choose three umbrellas U 1 , U 2 , and U 3 of v intersected by f , and choose one intersection of f with each chosen umbrella. Let U * (v) denote the intersection of an arbitrarily small neighborhood with U 1 ∪ U 2 ∪ U 3 , whose intersection with G consists of v and ends of edges incident to v. We let f i denote the chosen intersection of f with Let W denote the chosen facial boundary walk of f and let W = ω 1 ω 2 ω 3 for closed walks ω 1 , ω 2 , and ω 3 , as in Figure 7: ω 1 is a closed walk beginning by traversing edge end 6 as indicated, and ending at v after traversing the edge end 1 as indicated; ω 2 is a closed walk beginning by traversing the edge end 2 as indicated, and ending at v after traversing the edge end 3 as indicated; ω 3 is a closed walk beginning by traversing the edge end 4 as indicated, and ending at v after traversing the edge end 5 as indicated. We alter the order of ω 1 , ω 2 and ω 3 to produce a new facial boundary walk W ′ . We let W ′ = ω 1 ω 3 ω 2 , and let the set of facial boundary walks of G → P be W ′ = ({W ′ } ∪ W) \ {W }. The reordering of ω 1 , ω 2 , and ω 3 to produce W ′ is described in Figure 7.
To see that the dual graphs of G → P and G → P ′ are identical, note that the darts of W and the darts of W ′ are darts corresponding to the same edges of G. Therefore, the incidence of faces and edges of G → P is the same as that of G → P ′ . It follows that the dual graphs of G → P and G → P ′ are identical, and that P ′ is also face connected.
To see that P ′ has the same orientability as P , note that c(W ′ ) = c(W ), and so → dim(W ′ ) = → dim(W). Lemma 3.1 allows us to conclude that P ′ is orientable if and only if P is orientable.
To see that the three chosen umbrellas of the vertex v have been merged into one, we will define an auxiliary graph that captures the incidence of certain regions of G → P with G. Let We define a graph H whose vertex set is {f 1 , f 2 , f 3 , R 1 , R 2 , R 3 }, and with six edges {e 1 , e 2 , e 3 , e 4 , e 5 , e 6 }. Consider Figure 8; we define the edge e i to be incident to f j and R k if the edge end i is common to the boundaries of f j and R k . Note that H consists of three disjoint 2-cycles.
Because of the reordering of ω 1 , ω 2 , ω 3 to produce W ′ = ω 1 ω 3 ω 2 , the new facial boundary walk W ′ traverses the following ordered pairs of edge ends as as W ′ enters and exits U * (v): 1 and then 4, 5 and then 2, 3 and then 6, and in this order. Let f ′ be the face of G → P ′ bounded by the facial boundary walk W ′ . We define f ′ 1 , f ′ 2 , f ′ 3 to be the regions of f ′ ∩ U * (v) bounded by edge ends 1 and 4, 5 and 2, 3 and 6, respectively.
The graph H is drawn in gray. Each edge e i is drawn transversely crossing the edge end i.
Note that the process of producing W ′ does not affect the boundaries of R 1 , R 2 and R 3 . We define another, analogous graph H ′ for G → P ′ whose vertex set is It is easy to verify that this graph is a single cycle. We conclude that there are two fewer umbrellas of v in G → P ′ .

Applications of the surgeries to our embeddings
It is not difficult to show (see the proof of [15,Theorem 1.2]) that for a closed face-connected pseudosurface P with h handles, c crosscaps, and p pinches the Euler characteristic of P is Since, in a self-dual proper embedding G → P it must be true that |V (G)| = |F (G → P )|, it follows that For K 4m,4n to have a self-dual embedding K 4m,4n → P in some pseudosurface P , it follows from Equations 1 and 2 that We now apply the surgeries described in Theorems 3.1 and 3.2 to embeddings of K 4m,4n given in Section 2.2. Depending on m and n, there are many different face-connected pseudosurfaces whose Euler characteristic is 8m + 8n − 16mn, and our surgeries and construction can be used to find self-dual embeddings in many, but not all of them. Recall our choice of A = {a 0 , . . . , a 4m−1 } and B = {b 0 , . . . , b 4n−1 } denoting the bipartition sets of vertices of K 4m,4n . We first treat the case for which the surgically-produced pseudosurface is orientable.
Theorem 4.1. For each integer i such that 0 ≤ i ≤ 4m−1, let x i be an integer having the property that n − 2x i ≥ 1. For each integer j such that 0 ≤ j ≤ 4n − 1, let y j be an integer having the property hat m − 2y j ≥ 1. There exists a self-dual proper embedding of K 4m,4n in an orientable face-connected pseudosurface P with h handles and p pinches such that: the vertex a i has n − 2x i umbrellas and the vertex b j has m − 2y j umbrellas, and Proof. Let K 4m,4n → P denote the orientable self-dual embedding given in Theorem 2.1. Recall that by Theorem 2.1, K 4m,4n → P , as a doubly covered embedding, has the property that each vertex is covered by a facial boundary walk. The result now follows by applying Theorem 3.2 x i times to the vertex a i and y j times to the vertex b j . Each application of the surgery reduces the number of pinches by two while producing a pseudosurface with the same Euler characteristic as the pseudosurface to which the surgery was applied. It follows from Equation 2 that the number of handles must increase by one each time the surgery is applied.
We now turn to finding all nonorientable pseudosurfaces for which we may produce a selfdual embedding of K 4m,4n using the construction given in Theorem 2.1 and the surgery given in Theorem 3.1. For this, we will need a few more definitions and lemmas.
For surfaces S 1 and S 2 , we let the connected sum of S 1 and S 2 be the surface formed by removing a disc from S 1 , another disc from S 2 , and identifying the two remaining spaces along their boundaries.
To fully appreciate the proof of Lemma 4.1, one must be familiar with the presentation of a surfaces as a polygon with its edges pasted together. The reader will find a full treatment of this topic in [12,Chapter 1] or [13,Chapter 12].  Let P be a nonorientable face-connected pseudosurface with h handles, c crosscaps, and n pinchpoints p 1 , p 2 , . . ., p n such that p i has u i umbrellas. There is a homeomorphism φ : P → P ′ such that P ′ is a face-connected pseudosurface P ′ with c + 2 crosscaps, h − 1 handles, and n pinchpoints p ′ 1 , p ′ 2 , . . . p ′ n such that p ′ i has u i umbrellas.
Proof. We know P is obtained from a surface S with h handles and c crosscaps by identifying u i unique points to form each pinchpoint p i . For each p i , we let X i = {x k : 1 ≤ k ≤ u i } be the set of points thus identified. It follows that no two X i have any points in common. Let Q : S → P be the quotient map that for each i identifies the points of X i . Let S ′ be a surface with c + 2 crosscaps and h − 1 handles, and let f : S → S ′ be the homeomorphism whose existence is implied in 4.1. Let Q ′ : S ′ → P ′ be the quotient map that identifies the points of f (X i ) to p ′ i for all i. Let φ : P → P ′ be defined by φ = Q ′ •f •Q −1 . It is easy to see that φ is a homeomorphism mapping the pinchpoint p i to the pinchpoint p ′ i . Corollary 4.1. If G is properly embedded in P , then the homeomorphism φ : P → P ′ from Lemma 4.2 induces a proper embedding of G in P ′ such that the dual graphs of G → P and G → P ′ are isomorphic.
Proof. Since the complement of G in P is homeomorphic to a disjoint union of regions, each homeomorphic to a disc, it follows that since φ is a homeomorphism, the intersections of the boundaries of any faces are preserved. Since the vertices and edges of G form the boundaries of the faces of G → P , it follows that the incidence of faces and edges is preserved by φ. It follows that the dual graphs of G → P and G → P ′ are isomorphic.
Theorem 4.2. For each integer i such that 0 ≤ i ≤ 4m − 1, let x i be an integer having the property that n − x i ≥ 1. For each integer j such that 0 ≤ j ≤ 4n − 1, let y j be an integer satisfying m − y j ≥ 1. Suppose further that at least one of the x i or y j is nonzero. There is a self-dual embedding of K 4m,4n in the nonorientable face connected pseudosurface P ′ having h handles, c crosscaps, and p pinches such that: 1. 8m + 8n − 16mn = 2 − 2h − c − p, 2. the vertex a i has n − x i umbrellas and the vertex b j has m − y j umbrellas, and 3. p = i (n − x i − 1) + j (m − y j − 1).

Proof.
Consider the orientable self-dual embedding of K 4m,4n → P given in Theorem 2.1. Assume without loss of generality that x i * is nonzero. Applying the surgery given in Theorem 3.1 x i times at the vertex a i and y j times at the vertex b j , we see in light of Corollary 3.1, since the surgery was applied at least once at a i * , that we have produced a nonorientable self-dual embedding in a pseudosurface with the pinchpoint vertices having the desired number of umbrellas. Each application of the surgery reduces the number pinches by 1 while producing a pseudosurface with the same Euler characteristic as the one to which the surgery was applied. Invoking Lemma 4.2 and Corollary 4.1 enough times, we can find a self-dual embedding of K 4m,4n in a nonorientable pseudosurface with the desired number of handles and crosscaps; we exchange two crosscaps for a handle (or vice versa) each time.
In light of Lemma 4.2 we may conclude that there is only one nonorientable face-connected pseudosurface of a given Euler characteristic with a specific number of umbrellas at each pinchpoint. Theorem 4.2 helps us classify exactly which of these pseudosurfaces admits a self-dual embedding of K 4m,4n : every pseudosurface P that satisfies χ(P ) = 8m + 8n − 16mn and has an admissible number of pinchpoints and umbrellas admits a self-dual embedding of K 4m,4n except for the unique pseudosurface of Euler characteristic 8m + 8n − 16mn that has exactly 4m pinchpoints with n umbrellas and exactly 4n pinchpoints with m umbrellas. In other words, Theorem 4.2 does not guarantee the existence of a full self-dual embedding of K 4m,4n in a nonorientable pseudosurface.