On local operations that preserve symmetries and on preserving polyhedrality of maps

We prove that local operations that preserve all symmetries, as e.g. dual, truncation, medial, or join, as well as local operations that are only guaranteed to preserve all orientation-preserving symmetries, as e.g. gyro or snub, preserve the polyhedrality of simple maps. This generalizes a result by Mohar proving this for the operation dual. We give the proof based on an abstract characterization of these operations, prove that the operations are well defined, and also demonstrate the close connection between these operations and Delaney-Dress symbols.


Introduction
Symmetry-preserving operations on polyhedra have been studied for a very long time. They were first applied in ancient Greece. Some of the Archimedean solids can be obtained from Platonic solids by applying the operation which was later called truncation by Kepler. Over the centuries, polyhedra and specific operations on them have been studied extensively [3,11,12,18,22]. However, a general definition of the concept local symmetry-preserving operation and a systematic way of describing such operations was only presented in 2017 [2]. This description covers a large class of operations on maps, including all well-known symmetry-preserving operations such as truncation, dual, or those operations known as achiral Goldberg-Coxeter operations [4,5]. Goldberg-Coxeter operations were in fact introduced by Caspar and Klug [4] and can be used to construct all fullerenes or certain viruses with icosahedral symmetry. In addition to these local symmetry-preserving operations (lsp-operations), which preserve all the symmetries of a map, there are also operations that are only guaranteed to preserve the orientation-preserving symmetries. Well-known examples of such operations are snub and gyro [23], or the chiral Goldberg-Coxeter operations. In [2], a general description of such local orientation-preserving symmetry-preserving operations (lopsp-operations) was also presented. The very general way of describing lsp-and lopsp-operations in [2] allows to tackle various problems from a more abstract perspective, and also allows to prove general theorems about the whole class of operations instead of considering each operation separately. In this paper we will use the new description to prove that all those operations (e.g. dual, medial, truncation, snub, . . . ) preserve polyhedrality of maps i.e., if an lsp-or lopsp-operation is applied to a simple 3-connected map of face-width at least 3, then the result is also simple and 3-connected and it has face-width at least 3.
As the description in [2] was aimed at a broader audience than just mathematicians, the approach was described in a more intuitive way. In that article an operation is defined as a triangle 'cut' out of a simple periodic 3-connected tiling, and it is applied by gluing copies of that triangle into the barycentric subdivision of a map. Another way of looking at it is that the faces of the barycentric subdivision, which are triangles, are further subdivided into smaller triangles. This is done in a way that the subdivisions of the faces of the barycentric subdivision are identical or mirror images of each other, or -in case only orientation preserving symmetries must be preserved -in a way that each pair of two triangular faces of the subdivision that share the same edge as well as the same face of the map is subdivided in the same way. In the remainder of this text we will give the conditions for these subdivisions that guarantee that the result is the barycentric subdivision of another map -the result of the operation. An example of an lsp-operation and its application is shown in Figure 1. In this article, we will give the more direct definition based on Delaney-Dress symbols that forms the base of this approach and show the connection to the original description. We will also show that for every lsp-operation there is an equivalent lopsp-operation, i.e. a lopsp-operation that has the same result as the lsp-operation when applied to a map.
In [2], it is proved that the result of applying an lsp-operation to a polyhedron -that is: a simple 3-connected map embedded in the plane [17] -is also a polyhedron. In [14] this result is also announced for all lopsp-operations. We will modify some concepts that are used in that paper, but due to some serious problems in that paper we will not use the results given there.
Originally, lsp-as well as lopsp-operations were only defined for simple plane maps because of their origin in the study of polyhedra. However, there is no mathematical reason why these definitions should not be applied to maps with multiple edges or loops and embeddings of higher genus. The question then arises in how far we can extend the theorem for 3-connected simple plane maps to 3-connected maps of higher genus.
In general, lopsp-operations do not necessarily preserve 3-connectivity for maps that are not plane. This is obvious for maps with faces of size 1 or 2, but it is also true for simple maps in general, even if we require the result to be simple. The most striking example of a local symmetry-preserving operation that can turn 3-connected maps into (even simple) maps with lower connectivity is dual. In [1] it is proven that for any k ≥ 1, there exist embeddings of k-connected simple maps M so that the dual M * is simple and has a 1-cut.
However, even dual always preserves 3-connectivity in simple maps of face-width at least three, as proven in [20]. In Definition 4.1 and Definition 4.8 we will define ck-maps and ck-operations. A map is ck if it is k-connected, it has face-width at least k, and all of its faces have size at least k. In this paper we will prove the general Theorem 4.9 from which the following key result is a corollary. The result in [20]  This theorem is most interesting and relevant for k = 3. This has two reasons. Firstly, the set of c3-operations contains all well-known and intensely studied operations. Lspoperations that are not c3-lsp-operations were not even included in the original definition of lsp-operations [2]. Secondly, c3-maps, which are in fact simple embedded 3-connected maps of face-width at least three, have some very interesting properties. These maps are also known as polyhedral maps or polyhedral embeddings [20]. They can be defined equivalently as simple maps where every facial walk is a simple cycle and any two faces are either disjoint or their intersection consists of only one vertex or one edge. As the name suggests, polyhedral maps are a generalisation of polyhedra to surfaces of higher genus. It turns out that the key property that these operations preserve is not 3-connectivity but polyhedrality. This property is equivalent to being simple and 3-connected in the plane, but only in the plane. The main result of this article follows immediately from Corollary 1.1: If M is a polyhedral map and O is a c3-lspor c3-lopsp-operation, then O(M ) is also a polyhedral map (Theorem 4.10).
In Section 2 we give the definitions of the terminology we will use in this text. It starts with some basic concepts and then the definitions of lsp-and lopsp-operations are given. There is some freedom in the way that lopsp-operations are applied. However, in Section 3 we will prove that the result of applying a lopsp-operation is independent of the choices that are made in its application. Section 4 holds the main results of this paper: We prove a general result that implies that all lopsp-operations preserve polyhedrality of maps. To show that the definition of lopsp-operations we give is equivalent to the original definition in [2], we explore the strong connection between lsp-and lopsp-operations and tilings in Section 5. can be embedded such that it has genus 0. A plane map is one specific genus 0 embedding of a graph.
Let M be a map and G ′ a subgraph of the underlying graph of M . The map M ′ which is the graph G ′ with the embedding induced by that of M is called a submap of M .
To formalize when a vertex or edge is 'in' a face of one of its submaps we will now define what a bridge for a submap M ′ of M is. There are two kinds of bridges: Then the submap with vertex set V ′ C and edge set E ′ C is a bridge. If a bridge has an edge that is between two edges e and e ′ so that e −1 and e ′ form an angle in a face of M ′ , then the bridge is in that face. All the vertices and edges of the bridge are also said to be in the face. The boundary ∂f of a face f is the submap of M consisting of all the vertices and edges in the facial walk of f . A vertex or edge of M is in the interior of a face of M ′ if it is in that face and it is not in the boundary.
If a bridge is in more than one face, we say that those faces are bridged. A face that is not bridged is called simple.
Let f be a simple face of M ′ . We will define the internal component of f as follows. Start with the submap N of M that consists of the boundary of f together with all bridges in f . Intuitively, we cut along the boundary of f in N in such a way that the facial walk becomes a simple cycle. More formally, we replace every vertex v of N that appears k > 1 times in the facial walk of f by k pairwise different vertices v 1 , ..., v k . If both oriented edges associated with an edge of M ′ appear in the facial walk, this edge is also split into two different edges between different copies of its vertices. Let (x, v) and (v, y) be the oriented edges that form the angle in M ′ at the i-th occurrence of v. Then we define the rotational order (and also the neighbours) of v i to be the same as the rotational order around v in M , but restricted to the edges between (v, x) and (v, y). Of course some vertices may be replaced by their copies. The result of this is the internal component IC(f ) of f . An example of an internal component is illustrated in Figure 2. If IC(f ) is plane, we call f internally plane.
An important concept in the definition of lsp-and lopsp-operations is the barycentric subdivision of a map. It is obtained by subdividing every face into triangular faces, which we will call chambers. We will also use the barycentric subdivision to define contractible cycles and face-width in a combinatorial way. The barycentric subdivision B M of a map M is a map that has a unique vertex for every vertex, for every edge and for every face of M . We always assume that B M comes with the natural vertex-colouring that assigns colours 0, 1, and 2 to vertices that correspond to vertices, edges, and faces of M respectively. These colours correspond to their topological dimension. There are edges between vertices of colour 0 and 1 if the corresponding vertex and edge are incident. There are edges between vertices of colour 0 or 1 and colour 2 if the corresponding vertex or edge appears in the boundary of the corresponding face. There are no edges between vertices of the same colour. For i ∈ {0, 1, 2}, an edge is of colour i if it is not incident with a vertex of colour i. We will also refer to vertices and edges of colour i as i-vertices and i-edges. The rotational order of the edges adjacent to a vertex of colour 2 follows the order of the vertices and edges in the corresponding facial walk of M , and similarly for vertices of colour 0 and 1. This is illustrated in Figure 3. Every face of B M is a triangle. Note that in every figure in this text, colours are represented by colours in the order rgb, that is: a red is colour 0, green is colour 1, and black is colour 2. The edges of colour 1 are dashed and the edges of colour 2 are dotted, so that when looking at the figures printed in black and white it should still be clear which edges have which colour. With this rotation system, a short calculation of the Euler characteristic shows that gen(B M ) = gen(M ). If x is a face, edge or vertex of M , then to keep notation simple we will also write x for the corresponding vertex of B M .
Every face of B M is a triangle, with exactly one vertex and one edge of each colour. We call such a triangle a chamber. Two chambers are adjacent if they share an edge. In the literature, chambers are also called flags. The flag graph of M is the dual of B M , i.e. it is the 3-regular graph that has the chambers as its vertices, and there is an edge between two vertices if their corresponding chambers are adjacent. In some papers flags are defined as triples (v, e, f ) where v, e, and f are respectively a vertex, an edge, and a face such that v is a vertex of e and v and e are in face f [7,19]. We cannot use that approach here because with our general definition of a map there is no 1-to-1 correspondence between chambers and triples (v, e, f ). For example, an edge can have the same face on both sides so that there are multiple chambers with the same vertices. (ii) There are no edges between vertices of the same colour.
(iii) Every vertex of colour 1 has degree 4.
Proof. Let V G and E G be the sets of vertices of M with colours 0 and 1 respectively. Conditions (i) and (ii) imply that every face has exactly one vertex of each colour. It now follows from (iii) that a vertex e ∈ E G of colour 1 has two neighbours in V G and two neighbours f and g of colour 2. This induces an incidence relation on the vertex set V G and the edge set E G that defines a graph G. The rotation system of M induces a rotation system on G. Let N be the map that consists of G with this rotation system. It is not difficult to check that M = B N . In [2], lsp-and lopsp-operations are -following Goldberg [15] -defined in a geometric way as triangles 'cut' out of the barycentric subdivision of a 3-connected tiling of the plane, such that in case of lsp-operations the sides of the triangle are on symmetry axes of the tiling. In this article we give purely combinatorial definitions of lsp-and lopspoperations, similar to [14] and [13]. The definitions given here are equivalent to those in [2] when restricted to what we will later call c3-operations. The equivalence can be seen by applying operations as defined here to some special periodic tiling, but readers who want to see the equivalence already before starting on the main results of this paper and who want to have a deeper insight into the relation of operations and periodic tilings encoded by Delaney-Dress symbols, can find a direct proof without applications of the operations in Section 5.
Definition 2.2. Let O be a 2-connected plane map with vertex set V , together with a colouring c : V → {0, 1, 2}. One of the faces is called the outer face. This face contains three special vertices marked as v 0 , v 1 , and v 2 . We say that a vertex v has colour i if c(v) = i. This 3-coloured map O is a local symmetry preserving operation, lsp-operation for short, if the following properties hold: (1) Every inner face -i.e. every face that is not the outer face -is a triangle.
(2) There are no edges between vertices of the same colour, i.e. the colouring is proper.
(3) For each vertex that is not in the outer face: For each vertex v in the outer face, different from v 0 , v 1 , and v 2 : An example of an lsp-operation is shown in the middle of Figure 1. Just like for barycentric subdivisions we say that an edge is of colour i if it is not incident to a vertex of colour i. This is well-defined because of the second property.
Every inner face has exactly one vertex and one edge of each colour. We will refer to these triangular faces as chambers.
In the original paper [2] only operations that preserve 3-connectivity of polyhedra were discussed, so the result of the operation also had to have only vertices of degree at least 3. In [13] operations were also discussed that produce maps with 1or 2-cuts, but the restriction that vertices in the result should have degree at least 3 was kept. Our definition of lsp-operations is even more general. With this definition, the result of applying an lspoperation may have vertices of degree 1 or 2.

Application of an lsp-operation:
Let O be an lsp-operation and let M be a map. The operation is applied to M by first replacing for i ∈ {0, 1, 2} the i-edges of B M by copies of the part of the boundary of the outer face of O between v j and v k with i ̸ = j, k. The copy of v j is identified with the jvertex and the copy of v k with the k-vertex. Then -depending on the orientation -either a copy of O or a copy of the mirror image of O -which has the same underlying graph as O but the rotation system is the inverse of that of O -is glued into every face of the modified B M . Note that chambers of B M sharing an edge have different orientations. The boundary vertices are identified with their copies. This results in a 3-coloured triangulation. An example of the gluing -restricted to a single face -is given in Figure 1 As any symmetry group acts on the chamber system, lsp-operations preserve all the symmetries of a map. New symmetries can also occur. However, all known examples of 3-connected maps where lsp-operations can increase symmetry are maps of genus at least 1 or they are self-dual. It is an open question whether lsp-operations can increase symmetry in plane 3-connected maps (polyhedra) that are not self-dual.
There are also interesting operations such as gyro and snub that are only guaranteed to preserve the orientation-preserving symmetries of maps. These cannot be described by lspoperations. In the supplementary material of [2] and in [14], local orientation-preserving symmetry-preserving operations (lopsp-operations) are defined similarly to lsp-operations. The most important difference is that here the decoration is glued into double chambers instead of chambers. As with lsp-operations, we will give a more explicit definition of lopsp-operations that is not directly based on tilings.
There are some problems that arise in the original definition of lopsp-operations that do not appear for lsp-operations. With the original definition, it is possible to cut different patches out of a tiling that describe the same operation and must be shown to have the same result. That is why we define a lopsp-operation as a plane triangulation, similar to [14], and not as a quadrangle that we can glue directly into double chambers. Although this simplifies the definition of a lopsp-operation, the same problem comes back when it is described how the operation is applied.  (1) Every face is a triangle.
(2) There are no edges between vertices of the same colour, i.e. the colouring is proper.
(3) For each vertex v different from v 0 , v 1 , and v 2 : Again we say that an edge has colour i if it is not incident to a vertex of colour i and this is well-defined because of the second property. Note that the edges incident with a vertex have two different colours, and as every face is a triangle, these colours appear alternatingly in the cyclic order around the vertex. The requirement that O is 2-connected is mentioned in the beginning, but would in fact also follow from the other conditions. Again every face has exactly one vertex and one edge of each colour and will be referred to as a chamber. The dual of O will be referred to as the flag structure of O.

Application of a lopsp-operation:
For vertices v, v ′ in a path P we write P v,v ′ for the subpath of P from v to v ′ . As lopsp-operations are 2-connected, due to Menger's theorem there are two paths, one from v 0 to v 1 and one from v 0 to v 2 that have only v 0 in common. These paths together form a longer path P from v 1 to v 2 through v 0 . As a submap of O, P has a single face. In this facial walk only v 1 and v 2 occur once and all other vertices of P occur twice. We say that such a path P is a cut-path of O. Consider the internal component of the only face of submap P . This is the double chamber patch O P . It can be drawn in the plane, so that the two copies of P form the boundary of the outer face. Figure 4 shows this for the operation gyro. The result of the cutting is a 4-gon with corner vertices v 1 , v 2 , and two copies of v 0 , which we will denote as v 0,L and v 0,R . The flag structure of O P is the flag structure of O where the edges corresponding to edges of P are removed. The lopsp-operation is now applied by first replacing the edges of a double chamber map D M to form the map D M,P . An edge of colour 2 is replaced by a copy of P v0,v1 and an edge of colour 1 is replaced by a copy of P v0,v2 in a way that for i ∈ {0, 1, 2} a copy of v i is identified with a vertex of colour i.
Gluing copies of the double chamber patch O P into the faces of D M,P -identifying corresponding vertices in D M,P and the copies of double chamber patches -gives a coloured map BO P (M ). Note that the orientation inside a double chamber fixes how the different copies of v 0 have to be identified. Unlike with lsp-operations, we do not use mirrored copies of O. Figure 5 gives an example -restricted to one face -of this gluing. A side of a double chamber is a path in the boundary of the corresponding face of D M,P that is a copy of the path in O P between v 2 and v 0,L , between v 2 and v 0,R , or between v 0,L and v 0,R . A side is a 1-side if it is between copies of v 0 and v 2 and it is a 2-side if it is between two copies of v 0 . Proof. This follows immediately from Lemma 2.1.
As lsp-operations preserve all symmetries of a map, they also preserve the orientationpreserving symmetries, so one would expect that for every lsp-operation, there is a lopspoperation that has the same result when applied to any map. This observation allows to prove some properties of the result of applying lsp-or lopsp-operations only for lopspoperations. The result for lsp-operations can then be deduced from the corresponding lopsp-operation. Such an equivalent lopsp-operation can be obtained in the following way: Let  Proof. O lopsp is obviously a triangulation of the disc and there are no edges between vertices with the same colour. As O lopsp consists of two copies of O, glued along the boundary c, we can associate a unique vertex o( From the degree restrictions for lsp-operations we can now deduce the degree restrictions in the definition of lopsp-operations for O lopsp . It follows that O lopsp is a lopspoperation.
Choosing the cut-path in O lopsp that corresponds to the path from v 1 to v 2 through v 0 in c for the application of O lopsp shows immediately that the results of applying O and O lopsp are isomorphic: a double chamber is filled in the same way by O lopsp as two adjacent chambers are filled by O.

The path invariance of lopsp-operations
The cut-path chosen to apply an operation is far from unique, so there are many ways to apply a single lopsp-operation. In this section it is proved that although the ways in which the operation is applied differ, the result of applying a lopsp-operation to a map is independent of the chosen path. An essential tool in proving this are chamber flips, which simulate homotopic deformations.
Definition 3.1. Let P be a directed walk in a barycentric subdivision or lopsp-operation. For any two different vertices of a chamber C, there are two different simple paths P 0 , P 1 between these vertices in the boundary of C. If for i ∈ {0, 1} path P i occurs at a certain position in P , then a chamber flip of C (at this position) is the operation of replacing P i by P 1−i .
As a first tool we will discuss transformations of one path into another: Lemma 3.2. Let P, P ′ be two directed paths of the form P = P s R, P ′ = P s R ′ from x to y in a lopsp-operation T , so that R ′ R −1 is the facial walk of an internally plane face f in the submap of T consisting of the vertices and edges of P and P ′ .
Then there is a sequence of paths P = P 0 , P 1 , . . . , P k = P ′ so that for 1 ≤ i ≤ k path P i is obtained from P i−1 by a chamber flip and every vertex of P i is in P s or in the boundary or the interior of f . As chamber flips can be reversed, the same is true with the role of P and P ′ interchanged.
Proof. We will prove this by induction on the number |C | with C the set of chambers of T inside f . If |C | = 1, then R and R ′ are the two paths along the boundary of a chamber, so one can be transformed into the other by one chamber flip and we are done. Now assume that |C | ≥ 2. We prove that there are at least two chambers in C that have a connected intersection with ∂f that contains at least one edge: Let F f be the dual of T restricted to C and without edges that correspond to edges in ∂f . If T is the barycentric subdivision of a map then F f is part of the flag graph of that map. There are at least two chambers in C that contain an edge of ∂f . Assume that there is a chamber C such that C ∩ ∂f is disconnected. This chamber C splits the set C into two parts, i.e. the vertex corresponding to C is a cut-vertex of F f . In each component of F f \ C there is at least one chamber that shares an edge with ∂f . Let C 0 be a chamber that contains an edge of ∂f that has the largest distance d max to C along a path in F f . If this chamber has a disconnected intersection with ∂f , then its corresponding vertex is a cut-vertex of F f . This implies that there is a chamber that shares an edge with ∂f and has a larger distance to C than d max , which is in contradiction with the maximality of d max . Repeating this argument for the other component of F f \ C, it follows that in each of the two components there is a chamber that has a connected intersection with the facial walk ∂f that contains at least one edge.
Assume that one of these two chambers intersects ∂f in a single edge or in two edges of P or of P ′ . Then we can do a chamber flip to obtain either a path P 1 or P k−1 , so that we can apply induction to P 1 , P ′ or P, P k−1 and use that each chamber flip can be undone by a reverse chamber flip.
If the intersection of neither of the two chambers with ∂f is one or two edges of P or P ′ , then both intersections consist of one edge of P and one edge of P ′ . For one of the chambers, the shared vertex of those edges is the first vertex of R and R ′ . Applying a chamber flip replacing the edge of P , we get a path P 1 to which we can apply induction. Lemma 3.3. Let Q, Q ′ be two directed paths from x to y in a lopsp-operation, and z a vertex not contained in either of the paths.
Then there is a sequence of paths Q = Q 0 , Q 1 , . . . , Q k = Q ′ from x to y so that for 1 ≤ i ≤ k the path Q i is obtained from Q i−1 by a chamber flip and none of the paths contain z.
Proof. We will prove this by backwards induction on the number n of edges in the beginning of Q that Q and Q ′ have in common. Remember that for vertices If n = |Q ′ |, then Q = Q ′ , so assume that n < |Q ′ | and that the assumption is true for n ′ > n. Then there is a first vertex a in Q that is incident with an edge that is in Q ′ but not in Q. Let b be the next vertex after a in Q ′ that Q ′ shares with Q. We will show that Q can be transformed to Q ′ x,a Q ′ a,b Q b,y in the described way, so that we can apply induction We call the face of c containing z the exterior. Note that neither Q ′ x,a = Q x,a nor Q b,y intersects c in a vertex other than a or b. There are four possibilities for the position of Q x,a and Q b,y . These are depicted in Figure 6. If Q x,a or Q b,y are in the interior of c, we use them as part of the face boundary when applying Lemma 3.2, otherwise we do not. As Lemma 3.2 already allows to consider paths that start with a common part outside the face, we can choose P, P ′ from Lemma 3.2 in the following way: Note that in case Q x,a is outside c it forms the P s from Lemma 3.2, otherwise P s consists of a single vertex. In each case Lemma 3.2 can be applied to prove that Q can be transformed to Q ′ x,a Q ′ a,b Q b,y in the described way, and as the beginning of Q ′ x,a Q ′ a,b Q b,y has more than n edges in common with Q ′ , we can apply reverse induction.
Let M be a map, O a lopsp-operation with cut-path P and O P the corresponding double chamber patch. Recall that BO P (M ) is obtained by gluing copies of O P into D M . Therefore every vertex v in BO P (M ) is in at least one copy of O P . If v is in more than one copy, v corresponds to the same vertex of O in each of these copies. Similarly, every edge or face of BO P (M ) also corresponds to exactly one edge or face of O respectively. This allows us to define a surjective mapping π P , that maps every vertex, edge, and face of BO P (M ) to its corresponding vertex, edge, or face of O.
The mapping π P is not a bijection, but we can define a kind of inverse function π −1 P . It maps a set X of vertices, edges or faces in O to the set of all the vertices, edges or faces in BO P (M ) whose image under π P is in X. If we apply π −1 P to a single vertex, edge or face is a subset of vertices and edges of BO P (M ). If these form a connected graph, we interpret it as a map with the embedding induced by BO P (M ). The definition of BO P (M ) depends on P . We will now prove that the result of an operation is independent of P , so that we can define O(M ) for a lopsp-operation O. Proof. The idea of this proof is as follows. We define a submap BO P (M )| Q of BO P (M ) and prove that the underlying graph of this map is isomorphic as a graph to D M,Q . Then we prove that they are also isomorphic as maps, and that the internal component of each face Let e be an edge of D M , and let j be 1 if e has colour 2 and 2 if e has colour 1. Let P e be the copy of P vj ,v0 in BO P (M ) that replaced e. By Lemma 3.3 there is a series of paths P vj ,v0 = P 0 , . . . , P k = Q vj ,v0 from v j to v 0 in O, so that the path P i+1 is obtained from P i by a chamber flip of a chamber C i and none of v 0 , v 1 , v 2 occur as interior points of any of the paths. We define a sequence of paths P e = P e 0 , . . . , P e k in BO P (M ) with π P (P e i ) = P i for 0 ≤ i ≤ k. The path P e i+1 will be obtained from P e i by applying a chamber flip to a chamber C ∈ π −1 P (C i ). The chamber flips in O on the paths P i replace subpaths of one or two edges. In case of one edge it is clear that a corresponding chamber flip can be performed on P e i in BO P (M ). In case of two edges, we have to prove that the two corresponding edges of P e i are also contained in the same chamber. As P e i is a path, the two edges share one of their vertices, say v. By definition of the paths P i we get that . . , e k is the cyclic order of edges around v, then π P (e 1 ), π P (e 2 ), . . . , π P (e k ) is the cyclic order of the edges around the vertex π P (v). If a chamber flip is applied to the edges π P (e j ) and π P (e j+1 ) to go from P i to P i+1 , then we can apply a chamber flip to the edges e j and e j+1 to go from P e i to a new path P e i+1 . Thus our sequence of paths P e = P e 0 , . . . , P e k in BO P (M ) with π P (P e i ) = P i is defined for 0 ≤ i ≤ k and π P (P e k ) = Q vj ,v0 . We denote P e k as Q e . Note that Q e is isomorphic to Q vj ,v0 , not to Q. Let BO P (M )| Q be the map consisting of all the vertices and edges of BO P (M ) contained in Q e for some edge e. With the rotational orders induced by BO P (M ) we have that BO P (M )| Q is a submap of BO P (M ). First we prove that as (non-embedded) graphs, BO P (M )| Q and D M,Q are isomorphic.
Two paths Q e and Q e ′ can only intersect in their endpoints: Every other vertex v of Q e and Q e ′ satisfies π P (v) ̸ ∈ {v 0 , v 1 , v 2 }, which implies that v has only two incident edges that are mapped to edges in Q by π P . It follows that two paths of the form Q e are either disjoint -except possibly for their endpoints -or identical. We prove by induction that P e i and P e ′ i are disjoint (except for their endpoints) for all 0 ≤ i ≤ k and edges e and e ′ in D M . If e and e ′ are edges of a different colour this is trivial as at least one of their endpoints is different. Assume that e and e ′ have the same colour. By our previous argument it suffices to show that their first edge is different. For i = 0 this is clear. Assume that it is true for i − 1. Let ε i and ε ′ i be the first edges of P e i and P e ′ i respectively. We can assume that they are both incident with the same vertex x ∈ π −1 P ({v 0 , v 1 , v 2 }). The paths P e i and P e ′ i are obtained from P e i−1 and P e ′ i−1 by one chamber flip for each path. Either ε i = ε i−1 and ε ′ i = ε ′ i−1 , or the chamber flips replace ε i−1 and ε ′ i−1 by both their previous edges or both their next edges in the rotational order around x. As ε i−1 and ε ′ i−1 are different edges, ε i and ε ′ i are also different edges, which proves our statement. It follows that BO P (M )| Q and D M,Q are isomorphic as graphs. Next we prove that they are also isomorphic as maps.
Let m denote the total number of chamber flips necessary to transform first P v1,v0 to Q v1,v0 and then P v2,v0 to Q v2,v0 . With every face (that is: double chamber) D of D M and 0 ≤ i ≤ m we can now associate a closed walk W i that consists of the four paths P e1 i , P e2 i , P e3 i , P e4 i in BO P (M ) where e 1 , . . . , e 4 are the four edges of D, in the same order as they appear in D.
Claim: BO P (M )| Q is a submap of BO P (M ) that is isomorphic as a map to D M,Q and the internal component of each face is isomorphic to O Q .
Let C be the set of all chambers in BO P (M ), and let n be the number of chambers in O. We will define functions α i : C → Z (0 ≤ i ≤ m) with the following properties: (ii) For every chamber C in O: C ′ ∈π −1 P (C) α i (C ′ ) = 1 As a consequence of (ii) we have C∈C α i (C) = n. The walk W 0 is an internally plane facial walk of D M,P with an internal component that is isomorphic to O P . We define α 0 (C) = 1 if C is a chamber on the inside of W 0 and α 0 (C) = 0 if C is on the outside. As W 0 has exactly one copy of each chamber in O inside we get (ii) for α 0 . As α 0 only differs for neighbouring chambers if they share an edge of W 0 , and then in the way described by (i), we also get (i).
For i > 0 we define α i inductively. Let C be the chamber of O to which a chamber flip is applied when changing W i−1 to W i . These chamber flips occur in two places of W i−1 , and in fact in different directions. Two chambers C − , C + with π P (C − ) = π P (C + ) = C are involved, C − on the left of the cyclic walk and C + on the right. We now define α i (C − ) = α i−1 (C − ) − 1 and α i (C + ) = α i−1 (C + ) + 1. This is illustrated in Figure 7. As we once add one and once subtract one for two chambers with the same image under π P , (ii) is immediate. Property (i) can be checked easily by looking at α i for C − , C + , and the neighbouring chambers sharing an edge with them.
For i = 0, The function α i describes whether a chamber is inside or outside W i . For other i this is not always the case. If W i self-intersects the intuitive meaning of α i is less obvious.
For j = 1 or j = 2, the two edges of W i incident to the j-vertex x of D are always moved in the same direction by the chamber flips. This implies that M has a loop -every chamber that contains x is mapped to 0 or 1 by α m . The degree in W m of every vertex that is not in π −1 P ({v 0 , v 1 , v 2 }) is two, so we can follow W m from v 1 and from v 2 to the copies of v 0 to conclude that for each edge of W m , the two chambers C and C ′ containing it have α m (  We can also define the map π := π P as it is independent of the chosen path.

The effect of lsp-and lopsp-operations on polyhedrality
Polyhedral maps are simple maps that are 3-connected and have 'face-width' at least three. The face-width (or representativity) of a map is a measure of 'local planarity'. Embeddings of high face-width share certain properties with plane maps. We will define face-width in a combinatorial way, using barycentric subdivisions. It is not difficult to prove that the definition given here is equivalent to the definition in e.g. [20].  The condition that neither cuts with fewer than k vertices nor vertices with degree smaller than k may be present instead of just requiring the map to be k-connected is chosen in order to deal with small boundary cases. For example, a cycle is 2-connected, but its dual is a map with only two vertices so it is not 2-connected. Both a cycle and its dual are c2.
A polyhedral map is a simple, 3-connected map that has face-width at least three. Proof. It suffices to prove that every c3-map is simple and has at least four vertices. The rest of the statement is trivial when the definitions are written out. Let M be a c3-map. Facial loops and facial 2-cycles are excluded by the restrictions on face sizes and non-facial loops and non-facial 2-cycles imply either smaller cuts or a smaller face-width. Therefore M is simple. It has at least 4 vertices as it has minimum degree at least 3 and it is simple.
The reason why the term c3 is used instead of polyhedral in this article is that many results are proven for ck maps for general k ∈ {1, 2, 3}.
The following lemma characterises c2and c3-maps by a condition based on the chamber system. A 4-cycle in a barycentric subdivision is called trivial if it has a face that has no vertex or only a single colour-1 vertex in its interior.   be two edges between e f and the 0-vertex of c. This is a contradiction with the assumption that c is innermost. It follows that f c has a 0-vertex in its interior. As every 2-cycle has two well-defined sides, there is an innermost 2-cycle in the other face g c of c. Using the same arguments as for f c on that cycle we get that g c also has a 0-vertex in its interior. Every path in B M between the 0-vertices in the two faces using only 2-edges must pass through the 0-vertex of c. It follows that this vertex is a cutvertex of M , so that M is not c2.
(ii): Let M be a map and assume that M is not c3. If it is also not c2 we are done, so assume that M is c2. this cut consists of 2 vertices, a vertex and an edge or two edges. For each of the edges we can choose one of its incident vertices such that we find a cut-set consisting of 2 vertices, which is a contradiction with the 3-connectivity of M . Lemma 4.3 is very useful to determine whether a map is c2 or c3. It will often be used in the following lemmas and theorems. The main theorem of this last section is Theorem 4.9, which shows the equivalence of different definitions of ck-lopsp-operations and states that when applying ck-lopsp-operations with k ∈ {1, 2, 3} to certain maps, the result is ck. The most difficult part of its proof is captured in Theorem 4.5 for c2-maps and Theorem 4.7 for c3-maps.  is v 0 = t 1 , . . . , t k = v 1 and we denote one copy with t 1 , . . . , t k and the other with t ′ 1 , . . . , t ′ k , then -again due to minimality and as O has no loops -such an edge connects w.l.o.g. t i with t ′ i+1 for some 1 ≤ i < (k − 1). Considering two copies of O P sharing the copies of P v0,v1 , this gives a 4-cycle with v 1 in the interior. If c was trivial, then v 1 would be a 1-vertex adjacent to all 4 vertices on c -also t i and t ′ i -which contradicts the minimality of P . Both D 1 and D 2 contain x and y, so those vertices are on the boundary of both double chambers. Assume first that x and y are not both on copies of P v0,v1 or both on copies of P v0,v2 . Then D 1 and D 2 share their 1-vertex and their 2-vertex which implies a 2-cycle in B M , a contradiction. It follows that x and y are both on copies of P v0,v1 or both on copies of P v0,v2 . If they are in the same copy of P v0,v1 or P v0,v2 , then by Lemma 4.4 (i) an edge e 0 with the same vertices is also in that copy of P v0,v1 or P v0,v2 , so that O P contains a 2-cycle.  The last possibility is that x and y are in different copies of P v0,v1 or in different copies of P v0,v2 . As O does not contain loops we have π(x) ̸ = π(y). Applying Jordan's curve theorem to c we get that the edge e ′ 1 in D 2 with π(e 1 ) = π(e ′ 1 ) cannot exist -a contradiction (see Figure 9). The boundary ∂D of every double chamber D in S f is a cycle. As S f is plane, it follows with the Jordan curve theorem that c must cross ∂D an even number of times. By cross we mean that there is a subpath of c whose first and last vertex are on different sides of c, and whose other vertices are all in ∂D. Let D be a double chamber in S f that contains an edge of c in its interior. If c crosses ∂D 0 times, then c is contained in one copy of O P and we are done. If c crosses ∂D 4 times then there must be an edge outside of D that has both its vertices on ∂D. In this case Lemma 4.4 implies that there is a 2-cycle in O P and we are done. We can therefore assume that c crosses ∂D exactly twice. Note that every crossing is on a 1-side. Assume that the vertices of these crossings are on the same 1-side of D. If there would be only one edge of c in D this again leads to a 2-cycle in O P with Lemma 4.4. If there is no crossing in f it is clear that a chamber D ′ adjacent to D also has two crossings with c on the same 1-side. If f is in c that follows from the fact that f is on every 1-side and there must be at least two edges of c in a double chamber that has 2 crossings with c on the same 1-side. It follows that c is completely contained in D and D ′ , i.e. in two adjacent double chambers.
We can now assume that c has two crossings with ∂D that are on different 1-sides and not in f . As c crosses into the double chambers adjacent to D those must also have two crossings on different 1-sides. Repeating this argument we get that every double chamber in S f has two crossings with c on different 1-sides.
It follows that every double chamber in S f contains a subpath of c connecting vertices different from f on their two 1-sides. As there are at least three double chambers in S f and there must be at least one edge of c in each one, there are exactly three or four double chambers in S f . In each case there are at least two adjacent double chambers that contain only one edge of c. Let e 1 and e 2 be the only edges of c in two adjacent double chambers. As the vertices of e 1 and e 2 are on different 1-sides of O P the edges e 1 and e 2 are not in P . Therefore the edges π(e 1 ) and π(e 2 ) induce unique edges, that we will also denote with π(e 1 ) and π(e 2 ), in O P . If the vertices of P v2,v0 in O are -in this order -t 0 , t 1 , . . . , t s , then we will denote the vertices on the different 1-sides of O P with t 0 , t 1 , . . . , t s , resp. t ′ 0 , t ′ 1 , . . . , t ′ s . We have π(e 1 ) = {t i , t ′ j } and π(e 2 ) = {t j , t ′ k } with w.l.o.g. 0 < i < j ≤ s. Note that i ̸ = j as there are no loops in O. Due to the Jordan curve theorem applied to the cycle t 0 = t ′ 0 , t ′ 1 , . . . , t ′ j , t i , t i−1 , . . . , t 0 there is no edge {t m , t ′ l } in O P with m > i and l < j. As j > i and π(e 2 ) = {t j , t ′ k } is in O P , it follows that k > j. This situation is shown in Figure 10.
If there are four double chambers in S f we can repeat this argument on every pair of adjacent double chambers. With x 1 , x 2 , x 3 , x 4 the vertices of c in cyclic order and π(x a ) = t ia we get that i a < i a+1 for all 1 ≤ a ≤ 3 and i 4 < i 1 , so that by transitivity i 1 < i 1 , a contradiction. If there are only three double chambers in S f , then there must be two edges e 3 and e 4 of c in the same double chamber. The edges π(e 3 ) and π(e 4 ) form a path from t ′ i to t k . Such a path would have to cross both the edges π(e 1 ) and π(e 2 ), which is only possible if the path has at least three edges -it must contain t i or t ′ j and t j or t ′ kwhich is a contradiction. For the other implication we assume that O(M ) is not c3 but it is c2, and that there is no nontrivial 4-cycle in a patch of two adjacent copies of O P for a cut-path P . We will come to a contradiction by constructing such a 4-cycle. Let P be a cut-path in O of minimal length. We will refer to the copies of v 2 or v 0 in the double chamber patches as the corners of the double chamber patches. By Lemma 4.3 there is a nontrivial 4-cycle c in B O(M ) . Let X be a set of double chambers in D M of minimal size, so that the union of all double chamber patches for double chambers in X contains c. For simplicity we will also refer to the set of those double chamber patches as X. The fact that c has four edges implies that 1 ≤ |X| ≤ 4. If |X| = 1 then c can be thought of as a 4-cycle in O P , which we assumed does not exist, so |X| > 1. We make the following observations: (i) Every double chamber in X shares at least two of its corners with other elements of X: As |X| > 1 the cycle c 'enters' and 'leaves' any double chamber D ∈ X in two different vertices. Both of these vertices must be in the intersection of D with the other double chambers of X. It follows from (i) and (ii) that if |X| = 2 then c is a 4-cycle in two adjacent copies of O P , so |X| > 2. We will now prove that there is a cycle of length 4 in D M that contains at least one edge of each double chamber in X. We call such a cycle a saturating 4-cycle.
Assume first that every double chamber in X shares the same 2-vertex. With (i), (ii), and (iii), it follows easily that X consists of all the three or four double chambers corresponding to one face of M . Lemma 4.6 now implies that O(M ) is not c2 or that there is a 4-cycle in two adjacent copies of O P . Both are contradictions. Now assume that there is a 2-vertex f of D M such that there is only one edge e of c in the union of all the double chamber patches of X with 2-vertex f . The edge e has both its vertices on the same 2-side. As at least one of its vertices is not a corner, both double chambers sharing that 2-side are in X. This is a contradiction with (iii).
It follows that there are two 2-vertices f and g in D M such that the unions X f and X g of double chambers patches in X with 2-vertex f and g respectively each contain exactly two edges of c. With (i), (ii), and (iii) it follows that there are 0-vertices v and w of double chambers in X f and X g such that v, f, w, g is a saturating 4-cycle.
As M is c3, Lemma 4.3 implies that the saturating 4-cycle is the boundary of a double chamber, or two double chambers sharing the 1-vertex.
If e is the 1-vertex in these one or two double chambers, then c is contained in the set N e consisting of all six double chambers that share a side with a double chamber containing e. We say that the two double chambers with 1-vertex e are the central double chambers, and the four other double chambers in N e are the extremal double chambers. The three possible configurations of N e with respect to shared sides are shown in Figure 11. Using the fact that there are no nontrivial 4-cycles in B M it can easily be verified that no two vertices in Figure 11 represent the same vertex of D M .
We already proved that |X| ≥ 3 and that there are two edges of c in X f and two in X g . Therefore we can assume w.l.o.g. that X f consists of two double chambers. It follows from (iii) that these two double chambers are extremal double chambers. The two vertices of the edge e are in c. If the third vertex of c in X f is f , then with Lemma 4.4 and the fact that there are no 2-cycles in O P , it follows that the edges of c in X f are 1-edges of D M . In that case we can replace the two extremal double chambers in X by the central double chamber so X was not of minimal size, a contradiction. If the third vertex x of c in X f is not f , then the extremal double chambers share a side and x is on this side. Let e 1 and e 2 be the edges of c in these double chambers. The edge e 1 corresponds to an edge in O P from w.l.o.g. v 0,R to an internal vertex of P (v 0,L ),v2 and e 2 corresponds to an edge from v 0,L to P (v 0,R ),v2 . This would imply crossing edges in the plane map O P , a contradiction.
It follows that X f and X g each consist of only one double chamber, which by (i) must be the central double chamber. Then c is a nontrivial 4-cycle in a patch of two adjacent copies of O P , a contradiction.
For a lopsp-operation O, let T O be the tiling of the Euclidean plane obtained by applying O to the regular hexagonal tiling of the plane. We say T O is the associated tiling of O. With the definition for lopsp-operations from [2] this is the tiling from which O is defined. In Section 5 we will further explore the fundamental connection between lopsp-operations and tilings.
We will use the connectivity of the associated tiling of a lopsp-operation to define when an operation is ck. With Theorem 4.5 and Theorem 4.7 we will then prove an equivalent characterisation in Theorem 4.9 which does not depend on the associated tiling.  Assume now that O is not a c2-lopsp operation. Then there is a face f of size 1 or a 1-cut {x} in T O . In case of a 1-cut, at least one of the components of T O \ {x}, say C 0 , is finite. Let C denote a finite submap of the hexagonal skeleton that contains f , resp. C 0 together with the cut vertex x. Using Goldberg-Coxeter operations (see [2] or [4]) with sufficiently large parameters to construct large icosahedral fullerenes, we get a fullerene F , that is c3 and contains an isomorphic copy of C with the paths between the 0-vertices replaced by edges. Applying O to that fullerene, we get a submap S of O(F ) that has a face f ′ of size 1 or that is isomorphic to C 0 and where all vertices corresponding to vertices of C 0 have -except for the vertex x ′ corresponding to x -only neighbours in S. So f ′ or the vertex x ′ , which is a cut-vertex of O(F ), are contradictions to the assumption.
The case k = 3 is completely analogous, with also a 2-face and a 2-cut in the argument.    Figure 12: The operation truncation and the Delaney-Dress symbol encoding a tiling from which the operation can be obtained when the original definition is applied.

Connection to tilings
In a series of papers [8,9,10], Andreas Dress (in later papers together with coauthors) developed a finite symbol encoding the topology as well as the symmetry of periodic tilings. He attributed the idea to Matthew Delaney and called these symbols Delaney symbols. In later papers by other authors, these symbols are called Delaney-Dress symbols. In [6] and [10] Delaney-Dress symbols of periodic tilings of the Euclidean plane and the hyperbolic plane are characterized.
In this section we show that there is a very fundamental connection between l(op)spoperations and Delaney-Dress symbols and therefore to tilings. Recall that we defined the associated tiling T O of a lopsp-operation O as the tiling that is the result of applying O to the hexagonal tiling of the plane, i.e. the tiling with Schläfli symbol {6, 3}. We will find the same tiling in a different way using Delaney-Dress symbols, and we will see that from a mathematical point of view the choice of the tiling {6, 3} is quite arbitrary. The hexagonal tiling was chosen because it was also used in the original definition of lsp-operations in [2], where in turn it was chosen as a tribute to a paper by Goldberg [15]. By proving this connection it follows that our abstract combinatorial definitions are equivalent -in the 3-connected case -to the definitions of lsp-and lopsp-operations in [2].
As a topological definition of tilings falls outside the scope of this article, we will directly start with the combinatorial characterization described in [6,9,10]. We will sketch the connection to tilings, but for a detailed description we refer the reader to [6] or [10]. (1) D has finitely many elements Such Delaney-Dress symbols encode the combinatorial structure of periodic tilings of the Euclidean plane, together with a symmetry group acting on the tiling. If C (D, m 01 , m 02 , m 12 ) ̸ = 0, the tuple can also be a Delaney-Dress symbol, but then it encodes a periodic tiling of the hyperbolic plane (C < 0) or -in case additional divisibility rules are fulfilled -the sphere (C > 0) [6]. The elements of D are the orbits of chambers of the tiling under the symmetry group. An element C ∈ D with Cσ i = C represents an orbit of chambers with mirror symmetries of the tiling stabilizing the edges of colour i. If there are no C ∈ D with Cσ i = C, the symmetry group contains no pure reflections, but maybe sliding reflections. If there are no odd cycles, that is Cσ i1 . . . σ i k ̸ = C for odd k, all symmetries are orientation preserving. The maps m 01 and m 12 give information about the symmetry group of the tiling. Let {i, j, k} = {0, 1, 2}, i < j and for C ∈ D let r ij (C) = min{r | C(σ i σ j ) r = C}. Note that r ij is constant on ⟨σ i , σ j ⟩-orbits. If a ⟨σ i , σ j ⟩-orbit C ⟨σi,σj ⟩ contains no C ′ with C ′ σ i = C ′ or C ′ σ j = C ′ , then the vertices of colour k of the corresponding chambers in the tiling are centers of an f r -fold rotation with f r = m ij (C)/r ij (C). If an orbit C ⟨σi,σj ⟩ contains a C ′ with C ′ σ i = C ′ or C ′ σ j = C ′ , then with f m = 2m ij (C)/r ij (C) for f m > 1 the vertices of colour k of the chambers in orbit C are intersections of mirror axes with an angle of 360/f m degrees.
We will now associate a tuple (D O , m 01 , m 02 , m 12 ) with an lsp-or lopsp-operation O and prove that it is a Delaney-Dress symbol. In fact, it will be a Delaney-Dress symbol of the tiling O(T ) where T is the tiling with Schläfli symbol {6, 3}, i.e. the hexagonal tiling of the plane where every vertex has degree 3 and every face has 6 edges. Due to the relation between Delaney-Dress symbols and tilings as described in [6] and [10], this also shows the equivalence of the combinatorial definitions of lsp-and lopsp-operations defined here and the geometric ones given in [2]. There a l(op)sp-operation is described as a 'triangle' cut out of a tiling in such a way that certain conditions on the symmetry are satisfied.
One could replace the values 3 and 6 we will use for defining the mappings m ij by, for example, 4 and 4, and Theorem 5.3 would still be true. It would however be the Delaney-Dress symbol of the tiling that can be obtained by applying O to the square tiling of the plane, which is 4-regular and every face has 4 edges. By using other numbers, other tilings -even spherical or hyperbolic ones -could be used as source tilings. All of those tilings can be used to define l(op)sp-operations in the geometric way that was described in [2] for the hexagonal tiling.
Let O be an lsp-operation and let D O be the set of chambers of O. We define the action of Σ on D O by letting Cσ i = C ′ if C and C ′ share their i-edge, and Cσ i = C if the i-edge of C is in the outer face of O. For (i, j) ∈ {(0, 1), (0, 2), (1, 2)}, let v ij (C) be the vertex of chamber C that is not of colour i or j. We get: To find the Delaney-Dress symbol of the tiling obtained by applying O to {6, 3} we define m ij : D O → N as follows: Note that the requirements for the vertex degrees in an lsp-operation imply that for all C ∈ D O , the value m 02 (C) is 2.
We define D(O) = (D O , m 01 , m 02 , m 12 ) and call it the Delaney-Dress symbol corresponding to the lsp-operation O. This correspondence is illustrated for the operation truncation in Figure 12. Theorem 5.2 states that it is in fact a Delaney-Dress symbol of a tiling of the Euclidean plane.
By our previous remarks there is a 2-fold rotation around each copy of v 1 in that tiling, a 3-fold rotation around each copy of v 0 , and a 6-fold rotation around each copy of v 2 . There are also intersections of mirror axes with 90 • , 60 • , and 30 • angles at v 1 , v 0 , and v 2 respectively. This is the symmetry we expect when applying an lsp-operation to tiling {6, 3}. This is also the symmetry that is required to define an lsp-operation from a tiling with the geometric definition. Proof. We have to prove the properties in Theorem 5.1. The first two properties are obvious, so we will focus on the other two. Counting the number of chambers with a certain vertex and using the definition of m ij , we get that C 9 5 3 C 10 5 3 Figure 13: On the left, the double chamber patch of the lopsp-operation gyro is shown and on the right the corresponding Delaney-Dress symbol.
if v is an outer vertex different from v i for i = 0, 1, 2 Let n be the number of vertices (and equivalently edges) in the outer face. As every vertex of O has exactly one colour we get that: We will now prove the corresponding result for lopsp-operations. Let O be a lopspoperation and let D O be the set of chambers of O. We define the action of Σ on D O by letting Cσ i = C ′ if C and C ′ share their i-edge. For lopsp-operations there is no outer face, so r ij (C) is always deg(v ij ) 2 . We define m 01 , m 02 , m 12 : D O → N exactly as before: Again m 02 (C) = 2 for all C ∈ D O . We define D(O) = (D O , m 01 , m 02 , m 12 ) and in Theorem 5.3 we prove that it is a Delaney-Dress symbol. The operation gyro and its corresponding Delaney-Dress symbol are shown as an example in Figure 13. Once again, the tiling described by the Delaney-Dress symbol is the result of applying the operation to the hexagonal tiling of the plane. In Section 4 we named this tiling the associated tiling T O of O. There are 2-, 3-, and 6-fold rotations at the copies of v 1 , v 0 , and v 2 respectively. In lopsp-operations there is no chamber C such that Cσ i = C so there are no pure reflections encoded in the Delaney-Dress symbol. This is the symmetry required in the geometric definition of lopsp-operations. Proof. We prove the properties in Theorem 5.1. Again, the first two are obvious. Counting the number of chambers with a given vertex v and using the definition of m ij , we get that We can now compute C (D O , m 01 , m 02 , m 12 ): Proof. Mapping each chamber C lopsp of D(O lopsp ) onto the corresponding chamber C of D(O), we have (in the notation of [10]) a morphism between the symbols and in the notation of [6] a Delaney map f , that is: For all k ∈ {0, 1, 2}, (i, j) ∈ {(0, 1), (0, 2), (1, 2)}, and chambers C of D(O lopsp ) we have f (Cσ k ) = (f (C))σ k and m ij (C) = m ij (f (C)).
The existence of such a morphism guarantees (see [6,10]) that D(O) and D(O lopsp ) code combinatorially isomorphic tilings and that the tiling coded by D(O lopsp ) can be obtained from the tiling coded by D(O) by symmetry breaking -That is: modifying the tiling, so that the combinatorial structure is preserved, but some metric symmetries of the tiling are destroyed.

Future work
In the last section of [2] many open problems are described. They are sometimes just formulated for lsp-operations, but are often as relevant and interesting for lopsp-operations, so we refer the reader to [2]. A very interesting question is whether ambo is 'essentially' the only lsp-operation that can increase the symmetry of polyhedra, i.e. plane 3-connected maps. More specifically: Assume that for an lsp-operation O and a polyhedron M , the polyhedron O(M ) has more symmetries than M . Is M self-dual and can O be written as the product of ambo and other lsp-operations? For lopsp-operations this is certainly not true. For example, applying gyro to the tetrahedron gives the dodecahedron, which has a much larger symmetry group. Classifying lopsp-operations that can introduce new symmetries would be an interesting problem, but maybe even more difficult than solving the problem for lsp-operations.
We know that there is at least one lsp-operation (dual) that does not always preserve 3-connectivity for maps, if the face-width is at most two [1], so an obvious question is which other operations do not always preserve 3-connectivity. This was answered for lspoperations in [24], where the class of such operations, called edge-breaking operations, was characterized. Recently, these results have been extended to lopsp-operations. An article with the new results has been submitted [25].
Another problem mentioned in [2] -the generation of lsp-operations for a given inflation factor -has been solved [13]. Such an algorithm not only allows the generation of lsp-operations, but also the generation of polyhedra and other maps with some specific symmetry groups of the embedding. For generating lopsp-operations a program has been written very recently, but it has not been published yet.