Algorithmic Aspects of Regular Graph Covers
with Applications to Planar Graphs††thanks:
The conference version of this paper appeared in ICALP 2014.
The first three authors are supported by the ESF Eurogiga project GraDR as GAČR GIG/11/E023,
the fourth author by the ESF Eurogiga project GReGAS as the APVV project ESF-EC-0009-10 and by VEGA 1/0621/11.
The first author is also supported by the project Kontakt LH12095 and the second author by GAČR 14-14179S.
The second and the third authors are also supported by Charles University as GAUK 196213.
A graph covers a graph if there exists a locally bijective homomorphism from to . We deal with regular covers in which this locally bijective homomorphism is prescribed by an action of a subgroup of . Regular covers have many applications in constructions and studies of big objects all over mathematics and computer science.
We study computational aspects of regular covers that have not been addressed before. The decision problem RegularCover asks for two given graphs and whether regularly covers . When , this problem becomes Cayley graph recognition for which the complexity is still unresolved. Another special case arises for when it becomes the graph isomorphism problem. Therefore, we restrict ourselves to graph classes with polynomially solvable graph isomorphism.
Inspired by Negami, we apply the structural results used by Babai in the 1970’s to study automorphism groups of graphs. Our main result is the following FPT meta-algorithm: Let be a class of graphs such that the structure of automorphism groups of 3-connected graphs in is simple. Then we can solve RegularCover for -inputs in time where denotes the number of the edges of . As one example of , this meta-algorithm applies to planar graphs. In comparison, testing general graph covers is known to be NP-complete for planar inputs even for small fixed graphs such as or . Most of our results also apply to general graphs, in particular the complete structural understanding of regular covers for 2-cuts.
- 1 Introduction
- 2 Definitions and Preliminaries
- 3 Structural Properties of Atoms
- 4 Graph Reductions and Quotient Expansions
- 5 Meta-algorithm
- 6 Applying the Meta-algorithm to Planar Graphs
- 7 Concluding Remarks
The notion of covering originates in topology as a notion of local similarity of two topological surfaces. For instance, consider the unit circle and the real line. Globally, these two surfaces are not the same, they have different properties, different fundamental groups, etc. But when we restrict ourselves to a small part of the circle, it looks the same as a small part of the real line; more precisely the two surfaces are locally homeomorphic, and thus they share the local properties. The notion of covering formalizes this property of two surfaces being locally the same.
More precisely, suppose that we have two topological spaces: a big one and a small one . We say that covers if there exists a mapping called a covering projection which locally preserves the structure of . For instance, the mapping from the real line to the unit circle is a covering projection. The existence of a covering projection ensures that looks locally the same as ; see Figure 1a.
In this paper, we study coverings of graphs in a more restricting version called regular covering, for which the covering projection is described by an action of a group; see Section 2 for the formal definition. If regularly covers , then we say that is a quotient of .
1.1 Applications of Graph Coverings
Suppose that covers and we have some information about one of the objects. How much knowledge does translate to the other object? It turns out that quite a lot, and this makes covering a powerful technique with many diverse applications. The big advantage of regular coverings is that they can be efficiently described and many properties easily translate between the objects. We sketch some applications now.
Powerful Constructions. The reverse of covering called lifting can be applied to small objects in order to construct large objects of desired properties. For instance, the well-known Cayley graphs are large objects which can be described easily by a few elements of a group. Let be a Cayley graph generated by elements of a group . The vertices of correspond to the elements of and the edges are described by actions of on by left multiplication; each defines a permutation on and we put edges along the cycles of this permutation. See Figure 1b for an example. Cayley graphs were originally invented to study the structure of groups .
In the language of coverings, every Cayley graph with an involution-free generating set can be described as a lift of a one vertex graph with loops labeled . Regular covers can be viewed as a generalization of Cayley graphs where the small graph can contain more then one vertex. For example, the famous Petersen graph can be constructed as a lift of a two vertex graph , see Figure 2a. These two vertices are necessary as it is known that Petersen graph is not a Cayley graph. Figure 2b shows a simple construction  of the Hoffman-Singleton graph  which is a 7-regular graph with 50 vertices. Notice that from this construction it is immediately apparent that the Hoffman-Singleton graph contains many induced copies of the Petersen graph. (In fact, it contains 525 copies of it.)
The Petersen and the Hoffman-Singleton graphs are extremal graphs for the degree-diameter problem: Given integers and , find a maximal graph with diameter and degree . In general, the size of is not known. Many currently best constructions are obtained using the covering techniques .
Further applications employ the fact that nowhere-zero flows, vertex and edge colorings, eigenvalues and other graph invariants lift along a covering projection. In history, two main applications are the solution of the Heawood map coloring problem [39, 23] and construction of arbitrary large highly symmetrical graphs .
Models of Local Computation. These and similar constructions have many practical applications in designing highly efficient computer networks [15, 1, 6, 10, 11, 12, 24, 43] since these networks can be efficiently described/constructed and have many strong properties. In particular, networks based on covers of simple graphs allow fast parallelization of computation as described e.g. in [9, 2, 3].
Simplifying Objects. Regular covering can be also applied in the opposite way, to project big objects onto smaller ones while preserving some properties. One way is to represent a class of objects satisfying some properties as quotients of the universal object of this property. For instance, this was used in the study of arc-transitive cubic graphs , and the key point is that universal objects are much easier to work with. This idea is commonly used in fields such as Riemann surfaces  and theoretical physics .
1.2 Complexity Aspects
In the constructions we described, the covers are regular and satisfy additional algebraic properties. The reason is that regular covers are easier to describe. In this paper, we initiate the study of the computational complexity of regular covering.
|Input:||Connected graphs and .|
|Output:||Does regularly cover ?|
Relations to Covers. This problem is closely related to the complexity of general covering which was widely studied before. We try to understand how much the additional algebraic structure changes the computational complexity. Study of the complexity of general covers was pioneered by Bodlaender  in the context of networks of processors in parallel computing, and for fixed target graph was first asked by Abello et al. . The problem -Cover asks for an input graph whether it covers a fixed graph . The general complexity is still unresolved but papers [31, 20] show that it is NP-complete for every -regular graph where . (For a survey of the complexity results, see .)
The complexity results concerning graph covers are mostly NP-complete. In our impression, the additional algebraic structure of regular graph covers makes the problem easier, as shown by the following two contrasting results. The problem -Cover remains NP-complete for several small fixed graphs (such as , ) even for planar inputs . On the other hand, our main result is that the problem RegularCover is fixed-parameter tractable in the number of edges of for every planar graph and for every .
Two additional problems of finding lifts and quotients, closely related to RegularCover, are considered in Section 2.3.
Relations to Cayley Graphs and Graph Isomorphism. The notion of regular covers builds a bridge between two seemingly different problems. If the graph consists of a single vertex, it corresponds to recognizing Cayley graphs which is still open; a polynomial-time algorithm is known only for circulant graphs . When both graphs and have the same size, we get graph isomorphism testing. Our results are far from this but we believe that better understanding of regular covering can also shed new light on these famous problems.
Theoretical motivation for studying graph isomorphism is very similar to RegularCover. For practical instances, one can solve the isomorphism problem very efficiently using various heuristics. But polynomial-time algorithm working for all graph is not known and it is very desirable to understand the complexity of graph isomorphism. It is known that testing graph isomorphism is equivalent to testing isomorphism of general mathematical structures . The notion of isomorphism is widely used in mathematics when one wants to show that two seemingly different mathematical structures are the same. One proceeds by guessing a mapping and proving that this mapping is an isomorphism. The natural complexity question is whether there is a better way in which one algorithmically derives an isomorphism. Similarly, regular covering is a well-known mathematical notion which is algorithmically interesting and not understood.
Further, a regular covering is described by a semiregular subgroup of the automorphism group . Therefore it seems to be closely related to computation of since one should have a good understanding of this group first to solve the regular covering problem. The problem of computing automorphism groups is known to be closely related to graph isomorphism.
Homomorphisms and CSP. Since regular covering is a restricted locally bijective homomorphism, we give an overview of complexity results concerning homomorphisms and general coverings. Hell and Nešetřil  studied the problem -Hom which asks whether there exists a homomorphism between an input graph and a fixed graph . Their celebrated dichotomy result for simple graphs states that the problem -Hom is polynomially solvable if is bipartite and it is NP-complete otherwise. Homomorphisms can be described in the language of constraint satisfaction (CSP), and the famous dichotomy conjecture  claims that every CSP is either polynomially solvable, or NP-complete.
1.3 Our Results
Let be a class of connected multigraphs. By we denote the class of all regular quotients of graphs of (note that ). For instance for equal to planar graphs, the class is – according to the Negami’s Theorem  – the class of projective planar graphs. We consider four properties of , for formal definitions see Section 2:
The class is closed under taking subgraphs and under replacing connected components attached to 2-cuts by edges.
The graph isomorphism problem is solvable in polynomial time for and .
For a 3-connected graph , the group and all its semiregular subgroups can be computed in polynomial time. Here by semiregularity, we mean that the action of has no non-trivial stabilizers of the vertices.
Let and be 3-connected graphs of , possibly with colored and directed edges. Let the vertices of be further colored by , and let be equipped with a list of possible colors for each vertex (the coloring is not necessarily proper). We can test in polynomial time whether there exists a color-compatible isomorphism , i.e. an isomorphism such that the colors and orientations of edges are preserved and for every , we have . (The existence of such an isomorphism is denoted by .)
As we prove in Section 6, these four properties are tailored for the class of planar graphs. (But the proof of the property (P3) is non-trivial, based on the result of .) Negami’s Theorem  dealing with regular covers of planar graphs is one of the oldest results in topological graph theory; therefore we decided to start the study of computational complexity of RegularCover for planar graphs. Our algorithm, however, applies to a wider class of graphs.
We use the complexity notation which omits polynomial factors. Our main result is the following FPT meta-algorithm:
Let be a class of graphs satisfying (P0) to (P3). Then there is an FPT algorithm for RegularCover for -inputs in the parameter , running in time where is the number of edges of .
It is important that most of our results apply to general graphs. We wanted to generalize the result of Babai  which states that it is sufficient to solve graph isomorphism for 3-connected graphs. Our main goal was to understand how regular covering behaves with respect to vertex 1-cuts and 2-cuts. Concerning 1-cuts, regular covering behaves non-trivially only on the central block of , so they are easy to deal with. But we show that regular covering can behave highly complex on 2-cuts. From structural point of view, we give a complete description of this behaviour. Algorithmically, we solve computation only partially and we need several other assumptions to get an efficient algorithm.
Planar graphs are very important and also well studied in connection to coverings. Negami’s Theorem  dealing with regular covers of planar graphs is one of the oldest results in topological graph theory; therefore we decided to start the study of computational complexity of RegularCover for planar graphs. In particular, our theory applies to planar graphs since they satisfy (P0) to (P3).
For a planar graph , RegularCover can be solved in time .
Our Approach. We quickly sketch our approach. The meta-algorithm proceeds by a series of reductions replacing parts of the graphs by edges. These reductions are inspired by the approach of Negami  and turn out to follow the same lines as the reductions introduced by Babai for studying automorphism groups of planar graphs [4, 5]. Since the key properties of the automorphism groups are preserved by the reductions, computing automorphism groups can be reduced to computing them for 3-connected graphs . In [29, 28], this is used to compute automorphism groups of planar graphs since the automorphism groups of 3-connected planar graphs are the automorphism groups of tilings of the sphere, and are well-understood.
The RegularCover problem is more complicated, and we use the following novel approach. When the reductions reach a 3-connected graph, the natural next step is to compute all its quotients; there are polynomially many of them according to (P2). What remains is the most difficult part: To test for each quotient whether it corresponds to after unrolling the reductions. This process is called expanding and the issue here is that there may be exponentially many different ways to expand the graph, so we have to test in a clever way whether it is possible to reach . Our algorithm consists of several subroutines, most of which we indeed can perform in polynomial time. Only one subroutine (finding a certain “generalized matching”) we have not been able to solve in polynomial time.
This slow subroutine can be avoided in some cases:
If the -graph is 3-connected or if is odd, then the meta-algorithm of Theorem 1.1 can be modified to run in polynomial time.
Structure. This paper is organized as follows. In Section 2, we introduce the formal notation used in this paper. In Section 3, we introduce atoms which are the key objects of this paper. In Section 4, we describe structural properties of reductions via atoms, and expansions of constructed quotient graphs. In Section 5, we use these structural properties to create the meta-algorithm of Theorem 1.1. Finally, in Section 6 we deal with specific properties of planar graphs and show that the class of planar graphs satisfies (P0) to (P3). In Conclusions, we describe open problems and possible extensions of our results. See Section 2.4 for more detailed overview of the main steps.
2 Definitions and Preliminaries
A multigraph is a pair where is a set of vertices and is a multiset of edges. We denote by and by . The graph can possibly contain parallel edges and loops, and each loop at is incident twice with the vertex . (So it contributes by two to the degree of .) Each edge gives rise to two half-edges, one attached to and the other to . We denote by the collection of all half-edges. We denote by and clearly . As quotients, we sometime obtain graphs containing half-edges with free ends (missing the opposite half-edges).
We consider graphs with colored edges and also with three different edge types (directed edges, undirected edges and a special type called halvable edges). It might seem strange to consider such general objects. But when we apply reductions, we replace parts of the graph by edges and the colors encode isomorphism classes of replaced parts. This allows the algorithm to work with smaller reduced graphs and deduce some structure of the original large graph. So even if the input graphs and are simple, the more complicated multigraphs are naturally constructed.
2.1 Automorphisms and Groups
Automorphisms. We state the definitions in a very general setting of multigraphs and half-edges. An automorphism is fully described by a permutation preserving edges and incidences between half-edges and vertices. Thus, induces two permutations and connected together by the very natural property for every . In most of situations, we omit subscripts and simply use or . In addition, when we work with colored graphs, we require that an automorphism preserves the colors.
Groups. We assume that the reader is familiar with basic properties of groups; otherwise see . We denote groups by Greek letters as for instance . We use the following notation for standard groups:
for the symmetric group of all -element permutations,
for the cyclic group of integers modulo ,
for the dihedral group of the symmetries of a regular -gon, and
for the group of all even -element permutations.
Automorphism Groups. Groups are quite often studied in the context of group actions, since their origin is in studying symmetries of mathematical objects. A group acts on a set in the following way. Each permutes the elements of , and the action is described by a mapping usually satisfying further properties that arise naturally from the structure of .
In the language of graphs, an example of such an action is the group of all automorphisms of , denoted by . Each element acts on , permutes its vertices, edges and half-edges while it preserves edges and incidences between the half-edges and the vertices.
The orbit of a vertex is the set of all vertices , and the orbit of an edge is defined similarly as . The stabilizer of is the subgroup of all automorphisms which fix . An action is called semiregular if it has no non-trivial (i.e., non-identity) stabilizers of both vertices and half-edges. Note that the stabilizer of an edge in a semiregular action may be non-trivial, since it may contain an involution transposing the two half-edges. We say that a group is semiregular if the associated action is semiregular.
A graph covers a graph (or is a cover of ) if there exists a locally bijective homomorphism called a covering projection. A homomorphism from to is given by a mapping preserving edges and incidences between half-edges and vertices. It induces two mappings and such that for every . The property to be local bijective states that for every vertex the mapping restricted to the half-edges incident with is a bijection. Figure 3 contains two examples of graph covers. Again, we mostly omit subscripts and just write or . A fiber of a vertex is the set , i.e., the set of all vertices that are mapped to , and similarly for fibers over edges and half-edges.
The Unique Walk Lifting Property. Let be an edge which is not a loop. Then the set corresponds to a perfect matching between the fibers and . And if is a loop, then the set is a union of disjoint cycles which cover exactly . Figure 4 shows examples. In general for a subgraph of , the correspondence between and its preimages in is called lifting.
Let be a walk in . Then consists of copies of . Suppose that we fix as the start one vertex in the fiber . Due to the local bijectiveness of , there is exactly one edge incident with it which maps to , so is now uniquely determined, and so on for and the other vertices of . When we proceed in this way, the rest of the walk is determined. This important property of every covering is called the unique walk lifting property.
We adopt the standard assumption that both and are connected. Then as a simple corollary we get that all fibers of are of the same size. To see that, observe that a path in is lifted to disjoint paths in . For , consider a path between them. Then the paths in define a bijection between the fibers and . In other words, for some which is the size of each fiber, and we say that is a -fold cover of .
Covering Transformation Groups. Every covering projection defines a special subgroup of called covering transformation group . It consists of all automorphisms which preserve the fibers of , i.e., for every , the vertices and belong to one fiber. Consider the graphs from Figure 3. For the graph , we have and . And has but is trivial; and we note that it is often the case that a covering projection has only one fiber-preserving automorphism, the trivial one.
Now suppose that . Observe that a single choice of the image of one vertex fully determines . This follows from the unique walk lifting property. Let and consider some path connecting and in . This path corresponds to a path in . Now we lift and according to the unique walk lifting property, there exists a unique path which starts in . But since is an automorphism and it maps to , then has to be equal . In other words, we just proved that is semiregular.
Regular Coverings. We want to consider coverings which are highly symmetrical. For examples from Figure 3, the covering is more symmetric than . The size of is a good measure of symmetricity of the covering . Since is semiregular, it easily follows that for any -fold covering . A covering is regular if . In Figure 3, the covering is regular since , and the covering is not regular since .
We use the following definition of regular covering. Let be any semiregular subgroup of . It defines a graph called a regular quotient (or simply quotient) of as follows: The vertices of are the orbits of the action on , the half-edges of are the orbits of the action on . A vertex-orbit is incident with a half-edge-orbit if and only if the vertices of are incident with the half-edges of . (Because the action of is semiregular, each vertex of is incident with exactly one half-edge of , so this is well defined.) We naturally construct by mapping the vertices to its vertex-orbits and half-edges to its half-edge-orbits. Concerning an edge , it is mapped to an edge of if the two half-edges belong to different half-edge-orbits of . If they belong to the same half-edge-orbits, it corresponds to a standalone half-edge of .
2.3 Fundamental Complexity Properties of Coverings
We establish fundamental complexity properties of regular covering and also general covering. Our goal is to highlight similarities with the graph isomorphism problem. Also we discuss other variants of the RegularCover problem.
Belonging to NP. The general -Cover problem is clearly in NP since one can just test in polynomial time whether a given mapping is a locally bijective homomorphism. Not so obviously, the same holds for the RegularCover problem.
The problem RegularCover is in NP.
One just needs to use a suitable definition of regular covering. As stated above, regularly covers , if and only if there exists a semiregular subgroup of such that . As the certificate, we just give permutations, one for each element of , and the isomorphism between and . We can easily check whether these permutations define a group , and whether acts semiregularly on . Further, the given isomorphism allows to check whether the constructed is isomorphic to . Clearly, this certificate is polynomially large and can be verified in polynomial time.∎
One can prove even a stronger result:
For a mapping , we can test whether it is a regular covering in polynomial time.
Testing whether is a covering can clearly be done in polynomial time. It remains to test regularity. Choose an arbitrary spanning tree of . Since is a covering, then is a disjoint union of isomorphic copies of . We number the vertices of the fibers according to the spanning trees, i.e., such that . This induces a numbering of the half-edges of each fiber over a half-edge of , following the incidences between half-edges and vertices. For every half-edge , we define a permutation of taking to if there is a half-edge in such that the edge corresponds to .
It remains to test whether the size of the group generated by all , where , is of size exactly . From the theory of permutation groups, since is assumed to be connected, it follows that the action of is transitive. Therefore its size is at least , and the action is regular if and only if it is exactly .∎
The constructed permutations associated with are known in the literature  as permutation voltage assigments associated with .
Other Variants. In the RegularCover problem, the input gives two graphs and and we ask for an existence of a regular covering from to . There are two other reasonable variants of this problem we discuss now. The input can specify only one of the two graphs and ask for existence of the other graph of, say, a given size.
First, suppose that only is given and we ask whether a -fold cover of exists. This is called lifting and the answer is always positive. The theory of covering describes a technique called voltage assignment which can be applied to generate all -folds . We do not deal with lifting in this paper, but there are nevertheless many interesting computational questions with important applications. For instance, one can try to generate efficiently all lifts up to isomorphism; this is not trivial since different voltage assignments might lead to isomorphic graphs. Also, one may ask for existence of a lift with some additional properties.
The other variant gives only and asks for existence of a quotient which is regularly covered by and . This problem is NP-complete even for fixed , proved in a different language by Lubiw . Lubiw shows that testing existence of a fixed-point free involutory automorphism is NP-complete which is equivalent to existence of a half-quotient . We sketch hers reduction from -Sat. Each variable is represented by an even cycle attached to the rest of the graph. Each cycle has two possible regular quotients, either a cycle of half length (obtained by the rotation), or a path of half length with attached half-edges (obtained by a reflection through opposite edges). Each of these quotients represents one truth assignment of the corresponding variable. To distinguish variables, distinct gadgets are attached to the cycles. These variable gadgets are attached to clause gadgets. Naturally, one can construct a quotient of the clause gadget if and only if at least one literal is satisfied.
One should ask whether this reduction also implies NP-completeness for the RegularCover problem. Since the input gives also a graph , one can decode the assignment of the variables from it, and thus this reduction does not work. We conjecture that no similar reduction with a fixed can be constructed for RegularCover since we believe that for a fixed the problem of counting the number of regular coverings between and can be solved using polynomially-many instances of RegularCover. In complexity theory, it is believed that the counting version of no NP-complete problem satisfies this. Similar evidence was used by Mathon  to show that graph isomorphism is unlikely NP-complete, and as a work in progress we believe that a similar argument can be applied to RegularCover.
The results of this paper also show that the reduction of Lubiw cannot be modified for planar inputs . Our algorithmic and structural insights allow an efficient enumeration of all quotients of a given planar graph . On the other hand, the hardness result of Lubiw states that to solve the RegularCover problem in general, one has to work with both graphs and from beginning. Our algorithm starts only with and tries to match its quotients to only in the end. Nevertheless, some modifications in this directions, not necessary for planar graphs, would be possible.
2.4 Overview of the Main Steps
We now give a quick overview of the paper.
The main idea is the following. If the input graph is 3-connected, using our assumptions the RegularCover problem is trivially solvable. Otherwise, we proceed by a series of reductions, replacing parts of the graph by edges, essentially forgetting details of the graph. We end-up with a primitive graph which is either 3-connected, or very simple (a cycle or ). The reductions are done in such a way that no essential information of semiregular actions is lost.
Inspired by Negami  and Babai , we introduce in Section 3 the most important definition of an atom. Atoms are inclusion-minimal subgraphs which cannot be further simplified and are essentially 3-connected. Our strategy for reductions is to detect the atoms and replace them by edges. In this process, we remove details from the graph but preserve its overall structure. Our definition of atoms is quite technical, dealing with many details necessary for the next sections.
When the graph is not 3-connected, we consider its block-tree. The central block plays the key role in every regular covering projection. The reason is that the covering behaves non-trivially only on this central block; the remaining blocks are isomorphically preserved in . Therefore the atoms are defined with respect to the central block. We distinguish three types of atoms:
Proper atoms are inclusion-minimal subgraphs separated by a 2-cut inside a block.
Dipoles are formed by the sets of all parallel edges joining two vertices.
Block atoms are blocks which are leaves of the block-tree, or stars of all pendant edges attached to a vertex. The central block is never a block atom.
In Section 4, we deal with two transformations of graphs called reduction and expansion. The graph is reduced by replacing its atoms by edges; for proper atoms and dipoles inside the blocks, for block atoms by pendant edges. Further these edges are colored according to isomorphism classes of the atoms. This way the reduction omits unimportant details from the graph but its key structure is preserved. We apply a series of reductions till we obtain a graph called primitive which contains no atoms. We show that the reduction preserves essentially the automorphism group. More precisely, is a factor-group of ; the action of inside the atoms is lost by the factorization.
The other transformation called expansion is applied to the quotient graphs. The goal of expansion is to revert the reduction, so it replaces colored edges back by atoms. To do this, we have to understand how regular covering behaves with respect to atoms. Inspired by Negami , we show that each proper atom/dipole has three possible types of quotients that we call an edge-quotient, a loop-quotient and a half-quotient. The edge-quotient and the loop-quotient are uniquely determined but an atom may have many non-isomorphic half-quotients.
When the primitive graph is reached, all semiregular subgroups of are computed and for each one a quotient is constructed. Our goal is to understand all graphs to which can be expanded, as depicted in the following diagram:
The constructed quotients contain colored edges, loops and half-edges corresponding to atoms. Each half-edge in is created from a halvable edge if an automorphism of fixes this halvable edge and exchanges its endpoints. Roughly speaking it corresponds to cutting the edge in half. We show that every possible expansion of a quotient from can be constructed by replacing the colored edges by the edge-quotients of the atoms, the colored loops by the loop-quotients and the color half-edges by some choices of half-quotients. This gives the complete structural description of all graphs which can be reached from by expansion. Half-edges of can arise only in expansions of half-edges of .
In Section 5, we describe the meta-algorithm itself. From algorithmic point of view, the key difficulty arises from the fact that a graph can have exponentially many pairwise non-isomorphic expansions . Therefore we cannot test all of them and we proceed in the opposite way. We start with the graph and try to reach by a series of reductions. But here the reductions are non-deterministic since a part of the graph can correspond to many different subgraphs of . Therefore, we keep lists of possible correspondences when we replace atoms in . We proceed with the reductions and compute further lists using dynamic programming. There is only one slow subroutine which takes time which we describe in detail in Section 5.3.
In Section 6 we deal with specific properties of planar graphs and show that the meta-algorithm applies to them. It is a key observation that the RegularCover problem is trivially solvable for 3-connected inputs since the automorphism group is spherical; it is either cyclic, dihedral or a subgroup of one of the three special groups. One can just enumerate all quotients of and test graph isomorphism to . The reason is that 3-connected planar graphs and their quotients behave geometrically. On the other hand, dealing with general planar graphs is rather non-trivial, since the geometry of the sphere is lost and it requires all the theory built in this paper. We establish that planar graphs satisfy the conditions (P0) to (P3) where especially the proof for (P3) is not straightforward.
3 Structural Properties of Atoms
In this section, we introduce special inclusion-minimal subgraphs of called atoms. We show their structural properties such as that they behave nicely with respect to any covering projection.
3.1 Block-trees and Their Automorphisms
The block-tree of is a tree defined as follows. Consider all articulations in and all maximal -connected subgraphs which we call blocks (with bridge-edges also counted as blocks). The block-tree is the incidence graph between the articulations and the blocks. For an example, see Figure 6.
There is the following well-known connection between and :
Every automorphism induces an automorphisms .
First, observe that every automorphism of maps the articulations to the articulations and the blocks to the blocks which gives the induced mapping . It remains to show that is an automorphism of . Let be an articulation adjacent to a block in the tree. Then is contained in . Therefore is contained in and vice versa, which implies that is an automorphisms of the block-tree .∎
We note that there is no direct relation between the structure of and . First, may contain some additional automorphisms not induced by anything in . Second, several distinct automorphisms of may induce the same automorphism of . For example in Figure 6, and .
The Central Block. The center of a graph is a subset of the vertices which minimize the maximum distance to all vertices of the graph. For a tree, its center is either the central vertex or the central pair of vertices of a longest path, depending on the parity of its length. Every automorphism of a graph preserves its center.
If has a non-trivial semiregular automorphism, then has a central block.
For the block-tree , all leaves are blocks, so each longest path is of an even length. Therefore preserves the central vertex. The central vertex can be either a central articulation, or a central block. If the central vertex is an articulation , then every automorphism fixes which contradicts the assumptions.∎
In the following, we shall assume that contains a central block. We orient edges of the block-tree towards the central block; so the block-tree becomes rooted. A subtree of the block-tree is defined by any vertex different from the central block acting as root and by all its descendants.
Let be an articulation contained in the central block. By we denote the subtree attached to the central block at .
Let be a semiregular subgroup of . If and are two articulations of the central block and of the same orbit of , then . Moreover there is a unique which maps to .
Notice that either , or . Since and are in the same orbit of , there exists such that . Consequently . Suppose that there exist such that . Then is an automorphism of fixing . Since is semiregular, .∎
3.2 Definition and Basic Properties of Atoms
Let and be two distinct vertices of degree at least three joined by at least two parallel edges. Then the subgraph induced by and is called a dipole. Let be one block of , so is a 2-connected graph. Two vertices and form a 2-cut if is disconnected. We say that a 2-cut is non-trivial if and .
Let be a -cut and let be a component of . Then is uniquely determined by .
If is a component of , then has to be the set of all neighbors of in . Otherwise would not be 2-connected, or would not be a component of .∎
The Definition. We first define a set of subgraphs of which we call parts:
A block part is a subgraph non-isomorphic to induced by the blocks of a subtree of the block-tree.
A proper part is a subgraph of defined by a non-trivial 2-cut of a block not containing the central block. The subgraph consists of a connected component of together with and and all edges between and .
A dipole part is any dipole.
The inclusion-minimal elements of are called atoms. We distinguish block atoms, proper atoms and dipoles according to the type of the defining part. Block atoms are either pendant stars, or pendant blocks possibly with single pendant edges attached to it. Also each proper atom or dipole is a subgraph of a block. For an example, see Figure 7.
We use topological notation to denote the boundary and the interior of an atom . If is a dipole, we set . If is a proper or block atom, we put equal the set of vertices of which are incident with an edge not contained in . For the interior, we use the standard topological definition where we only remove the vertices , the edges adjacent to are kept.
Note that for any block atom , and for a proper atom or dipole . The interior of a dipole is a set of free edges. We note that dipoles are automatically atoms and they are exactly the atoms with no vertices in their interiors. Observe for a proper atom that the vertices of are exactly the vertices of the non-trivial 2-cut used in the definition of proper parts. Also the vertices of of a proper atom are never adjacent. Further, no block or proper atom contains parallel edges; otherwise a dipole would be its subgraph.
Properties of Atoms. Our goal is to replace atoms by edges, and so it is important to know that the atoms cannot overlap too much. The reader can see in Figure 7 that the atoms only share their boundaries. This is true in general, and we are going to prove it in two steps now.
The interiors of atoms are pairwise disjoint.
For contradiction, let and be two distinct atoms with non-empty intersections of and . First suppose that is a block item. Then corresponds to a subtree of the block-tree which is attached by an articulation to the rest of the graph. If is a block atom then it corresponds to some subtree, and we can derive that or . And if is proper atom or dipole, then it is a subgraph of a block, and thus subgraph of . In both cases, we get contradiction with minimality. Similarly, if one atom is a dipole, we can easily argue contradiction with minimality.
The last case is that both and are proper atoms. Since the interiors are connected and the boundaries are defined as neighbors of the interiors, it follows that both and are nonempty. We have two cases according to the sizes of these intersections depicted in Figure 8.
If , then is a 2-cut separating which contradicts minimality of and . And if, without loss of generality, , then there is no edge between and the remainder of the graph . Therefore, is separated by a 2-cut which again contradicts minimality of . We note that in both cases the constructed 2-cut is non-trivial since it is formed by vertices of non-trivial cuts and .∎
Next we show a stronger version of the previous lemma which states that two atoms can intersect only in their boundaries.
Let and be two atoms. Then .
We already know from Lemma 7 that . It remains to argue that, say, . If is a block atom, then is the articulation separating . No atom can contain this articulation as its interior. Similarly, if is a block atom, then has to be contained in the interior of or vice versa which contradicts minimality.
Let and . First we deal with dipoles. The situation where is a dipole is trivial. And if is a dipole with , then either which contradicts minimality of , or is not a 2-cut. It remains to deal with both and being proper atoms. Recall that in such a case is defined as neighbors of in , and that are neighbors of in .
The proof is illustrated in Figure 9. Suppose for contradiction that and let . Since has at least one neighbor in , then without loss of generality and . Since is a proper atom, the set is not a 2-cut, so there is another neighbor of in , which has to be equal . Symmetrically, has another neighbor in which is . So and . If and , the graph is (since the minimal degree of cut-vertices is three) which contradicts existence of 2-cuts and atoms. And if for example , then does not cut a subset of , so there has to a third neighbor of , which contradicts that cuts from the rest of the graph.∎
Connectivity of Atoms. We call a graph essentially 3-connected if it is a 3-connected graphs with possibly single pendant edges attached to it. For instance, every block atom is essentially 3-connected. A proper might not be essentially 3-connected. Let . We define as with the additional edge . Notice that the property (P0) ensures that belongs to . It is easy to see that is essentially 3-connected graph. Additionally, we put for a block atom or dipole.
Let be an essentially 3-connected graph, and we construct from by removing the single pendant edges of . Then is a subgroup of .
These pendant single edges behave like markers, giving a 2-partition of which has to preserve.∎
Further, if is of polynomial size, we can easily check which permutations preserve this 2-partition, and thus give . Also, similar relation holds for any group acting on and its subgroup preserving the 2-partition.
It is important that we can code the 2-partition by coloring the vertices of , and work with such colored 3-connected graph using (P2) and (P3).
3.3 Symmetry Types of Atoms
We distinguish three symmetry types of atoms which describe how symmetric each atom is. When such an atom is reduced, we replace it by an edge carrying the type. Therefore we have to use multigraphs with three edge types: halvable edges, undirected edges and directed edges. We consider only the automorphisms which preserve these edge types and indeed the orientation of directed edges.
Let be a proper atom or dipole with . Then we distinguish the following three symmetry types, see Figure 10:
The halvable atom. There exits an semiregular involutory automorphism which exchanges and . More precisely, the automorphism fixes no vertices and no edges with an exception of some halvable edges.
The symmetric atom. The atom is not halvable, but there exists an automorphism which exchanges and .
The asymmetric atom. The atom which is neither halvable nor symmetric.
If is a block atom, then it is by definition symmetric.
For a given dipole , it is possible to determine its type in polynomial time.
The type depends only on the quantity of distinguished types of the parallel edges. We have directed edges from to , directed edges from to , undirected edges and halvable edges. We call a dipole balanced if the number of directed edges in the both directions is the same. Observe that:
The dipole is halvable if and only if it is balanced and has an even number of undirected edges.
The dipole is symmetric if and only if it is balanced and has an odd number of undirected edges.
The dipole is asymmetric if and only if it is unbalanced.
This clearly can be tested in polynomial time.∎
For a given proper atom of satisfying (P2), it is possible to determine its type in polynomial time.
Let . Recall that is an essentially 3-connected graph. Let be the 3-connected graph created from by removing pendant edges, where existence of pendant edges is coded by colors of . Using (P3), we can check whether there is a color-preserving automorphism exchanging to as follows, see Figure 11. We take two copies of . In one copy, we color by a special color, and by another special color. In the other copy, we swap the colors of and . Using (P3) on these two copies, we can check whether there is an automorphism which exchanges and . If not, then is asymmetric. If yes, we check whether is symmetric or halvable.
Using (P2), we generate polynomially many semiregular involutions of order two acting on . For each semiregular involution, we check whether it transposes to , and whether it preserves the colors of coding pendant edges. If such a semiregular involution exists, then is halvable, otherwise it is just symmetric.∎
3.4 Automorphisms of Atoms
We start with a simple lemma which states how automorphisms behave with respect to atoms.
Let be an atom and let . Then the following holds:
The image is an atom isomorphic to . Further and .
If , then .
If , then .
(a) Every automorphism permutes the set of articulations and non-trivial 2-cuts. So separates from the rest of the graph. It follows that is an atom, since otherwise would not be an atom. And clearly preserves the boundaries and the interiors.
Therefore, for an automorphism of an atom , we require that . If a block or proper atom satisfying (P2), then we can compute according to Lemma 9 in polynomial time.
Projections of Atoms. Let be a semiregular subgroup of , which defines a regular covering projection . Negami [38, p. 166] investigated possible projections of proper atoms, and we investigate this question in more details. For a proper atom or a dipole with , we get one of the following three cases illustrated in Figure 12.
The atom is preserved in , meaning . Notice that is just a subgraph of . For a proper atom, it can happen that is adjacent, even through , as in Figure 12.
The interior of the atom is preserved and the vertices and are identified, i.e., and .
The covering projection is a -fold cover. There exists an involutory permutation in which exchanges and and preserves . The projection is a halved atom . This can happen only when is a halvable atom.
For every atom and every semiregular subgroup defining covering projection , one of the cases (C1), (C2) and (C3) happens. Moreover, for a block atom we have exclusively the case (C1).
For a block atom , Lemma 5 implies that , so the case (C1) happens. It remains to deal with being a proper atom or a dipole, and let . According to Lemma 12b every automorphism either preserves , or and are disjoint. If there exists a non-trivial which preserves , we get (C3); otherwise we get (C1) or (C2).
Let be a non-trivial automorphism of preserving . We know and by semiregularity, has to exchange and . Then the fiber containing and has to be of an even size, with being an involution reflecting copies of , and therefore the covering is a -fold cover. This proves (C3).
Suppose there is no non-trivial automorphism which preserves . The only difference between (C1) and (C2) is whether and are contained in one fiber of , or not. First suppose that for every non-trivial we get . Then no fiber contains more than one vertex of , and we get (C1), i.e, . And if there exists such that . By Lemma 12c, we get , so and belong to one fiber of , which gives (C2).∎
4 Graph Reductions and Quotient Expansions
We start with a quick overview. The reduction initiates with a graph and produces a sequence of graphs . To produce from , we find a collection of all atoms and replace each of them by an edge of the corresponding type. We stop at step when contains no further atoms, and we call such a graph primitive. We call this sequence of graphs starting with and ending with a primitive graph as the reduction series of .
Now suppose that is some quotient of . To revert the reductions applied to obtain , we revert the reduction series on and produce an expansion series of . We obtain semiregular subgroups such that . The entire process is depicted in the following diagram:
In this section, we describe structural properties of reductions and expansions. We study changes of automorphism groups done by reductions. Indeed, can differ from . But the reduction is done right and the important information of is preserved in which is key for expansions. The issue is that expansions are unlike reductions not uniquely determined. From , we can construct multiple . In this section, we characterize all possible ways how can be constructed from .
4.1 Reducing Graphs Using Atoms
The reduction produces a series of graphs . To construct from , we find the collection of all atoms and determine their types, using Lemma 11. We replace a block atom by a pendant edge of some color based at where . We replace each proper atom or dipole with by a new edge of some color and of one of the three edge types according to the type of . According to Lemma 8, the replaced parts for the atoms of are pairwise disjoint, so the reduction is well defined. We stop in the step when contains no atoms. We show in Lemma 16 that a primitive graph is either -connected, a cycle, or possibly with attached single pendant edges.
To be more precise, we consider graphs with colored vertices, colored edges and with three edge types. We say that two graphs and are isomorphic if there exists an isomorphism which preserves all colors and edge types, and we denote this by . We note that the results built in Section 3 transfers to colored graphs and colored atoms without any problems. Two atoms and are isomorphic if there exists an isomorphism which maps to . We obtain isomorphism classes for the set of all atoms such that and belong to the same class if and only if . To each color class, we assign one new color not yet used in the graph. When we replace the atoms of by edges, we color the edges according to the colors assigned to the isomorphism classes.
It remains to say that for an asymmetric atom we choose an arbitrary orientation, but consistently with for the entire isomorphism class. For an example of the reduction, see Figure 13.
The symmetry type of atoms depends on the types of edges the atom contains; see Figure 14 for an example. Also, the figure depicts a quotient of , and its expansions to and . The resulting quotients and contain half-edges because and fixes some halvable edges but contains no half-edges. This example shows that for reductions and expansions we need to consider half-edges even when the input and are simple graphs.
Properties of Reduction. Consider the groups and . There exists a natural homomorphism which we define as follows. Let . The graph is constructed from by replacing interiors of all atoms by colored edges. For the common vertices and edges of and , we define the image in exactly as in . If is an atom of , then according to Lemma 12a, is an atom isomorphic to . In , we replace the interiors of both and by the edges and of the same type and color. Therefore, we define .
More precisely for purpose of Section 4.2, we define on the half edges. Let and let and be the half-edges composing , and similarly let and be the half-edges composing . Then we define and .
The mapping satisfies the following:
The mapping is a group homomorphism.
The mapping is surjective.
Moreover, monomorphically embeds into .
For a semiregular subgroup of , the mapping is an isomorphism. Moreover, the subgroup remains semiregular.
(a) It is easy to see that each . The well-known Homomorphism Theorem states that is a homomorphism if and only if the kernel , i.e., the set of all such that , is a normal subgroup of . It is easy to see that the kernel has the following structure. If , it fixes everything except for the interiors of the atoms. Further, , so can non-trivially act only inside the interiors of the atoms.
Let and . We need to show that . Let be an atom. Then is an isomorphic atom. The composition clearly permutes the interior . Moreover, the part of the graph outside of interiors is fixed by the composition. Hence it belongs to and by the Homomorphism Theorem is a homomorphism.
(b) Let , we want to extend to such that . We just need to describe this extension on a single edge . If is an original edge of , there is nothing to extend. Suppose that was created in from an atom in . Then is an edge of the same color and the same type as , and therefore is constructed from an isomorphic atom of the same symmetry type. The automorphism prescribes the action on the boundary . We need to show that it is possible to define an action on consistently.
If is a block atom, then the both edges and are pendant, attached by articulations and . We just define using an isomorphism from to which takes to . It remains to deal with proper atoms and dipoles.
First suppose that is an asymmetric atom. Then by definition the orientation of and is consistent with respect to . Since , we define on according to one such isomorphism.
Secondly suppose that is symmetric or halvable. Let be an isomorphism of and . Either maps exactly as , and then we can use for defining . Or we compose with the automorphism of exchanging the two vertices of . (We know that such an automorphism exists since is not antisymmetric.)
(c) Let be an orbit of on edges representing atoms . We construct the extension as above, choosing any isomorphism from to , where , and properly the isomorphism to . Here, properly means that composition of all these isomorphisms is the identity on . Repeating this procedure for every orbit of , we determine an extension of defining a monomorphic embedding .
(d) We note that for any subgroup , the restricted mapping is a group homomorphism with . If is semiregular, then is trivial. The reason is that contains at least one atom , and the boundary is fixed by . Hence is an isomorphism.
For the semiregularity of , let be an automorphism of . Since is an isomorphism, there exists the unique such that . If fixes a vertex , then fixes as well, so it is the identity, and . And if only fixes an edge , then exchanges and . Since does not fix , then there is an atom replaced by in . Then is an involutary semiregular automorphism exchanging and , so is halvable. But then is a halvable edge, and thus can fix it.∎
The above statement is an example of a phenomenon known in permutation group theory. Interiors of atoms behave as blocks of imprimitivity in the action of . It is well-known that the kernel of the action on the imprimitivity blocks is a normal subgroup of . For the example of Figure 14, we get and . As a simple corollary, we get:
We already proved that .∎
Actually, one can prove much more, that . First, we describe the structure of .
The group is the direct product where is the point-wise stabilizer of in .
According to Lemma 7, the interiors of the atoms are pairwise disjoint, so acts independently on each interior. Thus we get as the direct product of actions on each interior which is precisely .∎
Alternatively, is isomorphic to the point-wise stabilizer of in . Let