These graphs have been considered by Heiko Theißen in Th for the computation of normalizers of subgroups of permutation groups, but our work does not build on his – partly because our hypothesis is more general and partly because his results have not been published except for in his PhD thesis.
But our motivation resembles Theißen’s and is mainly computational; for example we use orbital graphs in JPW for refinements of partitions in order to improve search algorithms that are based on partition backtrack methods. We are convinced that the application in JPW
is just the beginning and that huge computational benefits can be gained from a better understanding of these graphs in the future. In this article we classify those orbital graphs that are useless for computational purposes and we describe ways to detect these so-called “futile” graphs before they are even explicitly constructed. This way orbital graphs can be used most effectively in algorithms.
, in particular for orbits, point stabilizers etc., but we introduce everything that might not be standard. In Section 2 we discuss some basic theoretical results about orbital graphs that are probably well-known but that, to our knowledge, are mostly not contained in the existing literature. Then in Section 3 we prove specific new results that are motivated from a computational perspective and we finish with some open questions.
2 Basic Properties of orbital graphs
We begin by defining orbital graphs, by explaining some examples and by proving some basic properties. Our notation is standard, and we point out that by a proper digraph we mean a digraph that has at least one arc such that there its reverse arc is not in the graph. All digraphs considered here have no multiple arcs and no loops.
Let be a group of permutations on a set and let be distinct elements. Then the orbital graph of with base-pair is defined in the following way:
The vertex set is and the arc set is , where for all and we denote by the image of under the permutation .
Once and its action on are given, we denote the orbital graph of with base-pair by
One more general definition:
An isolated vertex of a digraph is a vertex with no arcs going into it or coming out of it.
For symmetric groups in their natural action we always obtain a complete self-paired digraph as orbital graph, independent of the base-pair.
Next we let and .
If we choose the base-pair , then this is the only arc in the orbital graph and the points are isolated.
The base-pair gives an orbital graph with arcs and , and we obtain a maximum number of arcs by choosing the base-pair . Then we have four arcs in total, namely and .
Let be a finite set, let and let be distinct. Let and let denote the set of arcs of .
Suppose that Hypothesis 2.3 holds. Then we have the following:
if and only if .
is self-paired if and only if some interchanges and .
is precisely the set of vertices of that are the starting point of some arc.
is precisely the set of vertices of that are the end point of some arc.
The number of arcs starting at is and the number of arcs going into is .
(i) If , then by definition is an arc in .
Conversely, suppose that is an arc in . Then there exists some such that . Hence the orbital graph generated by is the same as the orbital graph generated by .
By (i) coincides with if and only if the arc exists in , which happens if and only if
there exists some such that and .
(iii) Let and let be such that . Then is an arc with starting point .
Conversely, if is such that is an arc in , then there exists some such that and hence .
Similar arguments show (iv).
(v) We just calculate that the number of arcs starting at is
The number of arcs going into is, by the same reasoning,
(a) Parts (iii) and (v) of the lemma, together, give the total number of arcs in . The number of arcs starting at is exactly , so we obtain .
(b) In (ii) it is not true that must contain the transposition . A counterexample is provided by acting naturally on and its orbital graph with base-pair .
Suppose that Hypothesis 2.3 holds. Then the following are equivalent:
(a) has no isolated vertices.
If has some isolated vertex , then there is no arc starting at and no arc ending there. By Lemma 2.4 (ii) and (iii) it follows that . Hence (b) implies (a).
Conversely we suppose that (a) holds and we let . As is not isolated, it is the starting point of some arc or the end point. In the first case and in the second case by Lemma 2.4. It follows that as stated in (b). ∎
Suppose that Hypothesis 2.3 holds.
If acts transitively on , then has no isolated vertices. If is not transitive and , then is bipartite and has no isolated vertices.
The first statement follows from Lemma 2.6 because then .
The second statement follows from the same lemma together with Lemma 2.4 (iii) and (iv). ∎
Suppose that Hypothesis 2.3 holds. Then acts on as a group of graph automorphisms.
Of course acts faithfully on the set which is the vertex set of . For all vertices and all we write for the image of under in the original permutation action. Let .
If is an arc, then there exists some such that by definition of . Hence is an arc.
Conversely, if is an arc, then there exists some such that and hence is an arc.
As is a group, the induced maps are bijective and hence every induces a graph automorphism on . ∎
Suppose that Hypothesis 2.3 holds and let denote the connected component that contains . Then every connected component of that has size at least is isomorphic to .
Let denote an arbitrary connected component of of size at least and let be an arc in .
From the definition of orbital graphs let be such that . Then induces an automorphism on by Lemma 2.8 and it moves all arcs from to arcs in .
Conversely, induces an automorphism on that moves all arcs of into . Thus it follows that and are isomorphic as graphs. ∎
Lemma 2.10 shows how to generate a set of base-pairs which generate all orbital graphs for a group . Parts (i) and (ii) show to take a representative from each orbit of as the first element of the base-pair, and then part (iii) shows we must stabilize this point in , and take a representative from each orbit in this stabilizer for the second element of our base-pair. These base-pairs will allow us to analyse the set of orbital graphs of a group, before we construct any orbital graphs explicitly.
Let be a finite set and let .
Suppose that and . Then the set of orbital graphs of with base-pairs starting with is equal to the set of orbital graphs of with base-pairs starting with .
Suppose that and . Then the set of orbital graphs of with base-pairs starting with is disjoint from the set of orbital graphs of with base-pairs starting with .
Suppose that and that , . Let and . Then if and only if .
Let be such that . Then for all , it follows that . Conversely .
Suppose that the pairs and generate the same orbital graph of . Then by Lemma 2.4 (i) there is such that , which implies that . This proves the statement.
If , then and generate the same orbital graph. So by Lemma 2.4 (i) there is such that . This means that and therefore , which implies that . Conversely, if then there exists such that and hence .
After these preparatory results we can embark on the topic of usefulness of these graphs in algorithms.
3 Usefulness of orbital graphs in algorithms
Reasoning about arbitrary permutation groups is computationally extremely expensive. Therefore, Leon’s partition backtrack algorithm (see L ) replaces groups with the stabilizer of an ordered orbit partition during search. This can be seen as an approximation: A group G is a subgroup of the stabilizer of its ordered orbit partition in any supergroup of G. Using the intersection of the automorphism groups of all orbital graphs instead gives a smaller group and hence a faster algorithm – but there are exceptions where this approach does not give any advantage. This motivates the following definition:
Suppose that Hypothesis 2.3 holds and that is an ordered orbit partition of . We denote the stabilizer of in by and we emphasize that stabilizes every -orbit (i.e. every cell of the ordered partition ) as a set and that it acts as the full symmetric group on every orbit.
We say that the orbital graph is futile if and only if , in its natural action on , induces graph automorphisms on .
Just as a reminder:
A digraph is said to be a complete digraph if and only if its set of arcs is .
is called a complete bipartite digraph if and only if there exist pair-wise disjoint subsets of vertices such that is the disjoint union of (the “starting” vertices) and (the “end” vertices) and the set of arcs is exactly .
Our main theoretical result on this topic classifies futile orbital graphs. In particular, Corollary 3.6 shows how an orbital graph can be recognized as futile before it is even constructed. We note that the following result does not place any restrictions about the number of orbits of on . In particular there could be arbitrarily many isolated points in .
Suppose that Hypothesis 2.3 holds. Then is futile if and only if it has a unique connected component of size at least and moreover one of the following holds:
(a) is a complete digraph or
(b) is a complete bipartite digraph.
Let be an ordered orbit partition of . acts on the set of orbits of and it acts faithfully on the set of vertices of . Hence to answer if is futile or not, we only have to consider arcs in .
We split our proof into two cases depending on whether or not is a proper digraph.
Case 1: is a proper digraph.
We begin with the hypothesis that is futile.
(1) Suppose that are distinct and in the same -orbit. Then they are not on an arc. In particular .
As and are in the same -orbit, they lie in the same cell of the partition . It follows from the futility of that the transposition , which stabilizes , induces a graph automorphism on . Therefore neither nor is an arc. From this and the fact that it follows that . ∎
(2) Suppose that is on an arc. Then it is either a starting point or an end point, but not both.
This follows from Lemma 2.4 (iii) and (1). ∎
Let and , and let denote the set of isolated vertices of .
(3) . Moreover spans the unique connected component of of size at least , and this component is a complete bipartite digraph.
The first statement follows from (2). Moreover there are no arcs between vertices in or , respectively, by (1). We show that all elements of are on an arc with :
For all , we find the transposition , and it fixes point-wise by (1). The futility of implies that maps the arc to the arc . Now it follows that and hence the digraph spanned by is a complete bipartite digraph, and it is the unique connected component of size at least of . ∎
Conversely, we suppose that has a unique connected component of size at least and that this component is a complete bipartite digraph. We prove that is futile.
Let and denote the subsets of the vertex set of such that all arcs start at and end at . Let be the set of isolated vertices of , so that .
Now and the bipartite structure implies that even . Similarly . Therefore stabilizes the sets , and . We already know that permutes the vertices of faithfully, so now we look at arcs.
Let and let . Then , and there exists some such that . Since stabilizes the sets and , we see that and . The completeness property then implies that .
Conversely, if , then there exists some such that . Now and whence by completeness.
Hence is futile.
Case 2: is not a proper digraph, which means that it is self-paired.
We begin, once more, with the hypothesis that is futile. Let denote the connected component of that contains the base-pair and let be an arbitrary, non-isolated vertex.
We know that by Lemma 2.4 (iii) and (iv), because is self-paired. Since some arc starts or ends in , we also have that and hence are all in the same -orbit and hence in a common cell of the partition . In particular the transposition is contained in and because of the futility it induces an automorphism on .
Then implies that . This argument shows that is a complete digraph and that it is the
only connected component of size at least in .
We conversely suppose that has a unique connected component of size at least and that it is complete. Together with the definition of orbital graphs (and the fact that arcs always go both ways in the present case) this implies that spans the unique connected component of size at least and that the isolated vertices, viewed as elements of , are not contained in .
We know that acts faithfully on the vertex set of . Now let and let . We recall that is -invariant.
Then it follows as in Case 1, using the completeness, that if and only if . Consequently acts as a group of automorphisms on , i.e. is futile. ∎
We give an example in order to illustrate that futility of an orbital graph is not obvious and why further investigations into the usefulness of orbital graphs should be pursued.
We let and we look at the subgroup
. Let be the orbital graph for with base-pair . Then has the following shape:
On the vertices and we have a complete digraph, respectively, there is no arc between the sets and , and the points and are isolated. This might look like a futile graph, but according to the theorem it is not. Consider an ordered orbit partition of .
The group contains the transposition . This element interchanges the vertices and of and fixes , so this element does not induce an automorphism on . (Otherwise the arc would be mapped to the arc , which does not exist). This graph can be used to deduce, for example, that any element which swaps and must also swap with .
Hence does not act as a group of automorphisms on and we see that is, in fact, not futile.
It is important that we can detect futile graphs easily, without having to create them explicitly. We will now give a collection of Lemmas which allow futile orbital graphs to be detected using only information about orbits and stabilizers of a group, without explicit construction of entire orbital graphs.
Suppose that Hypothesis 2.3 holds and that , where is the set of isolated vertices of . Then is futile if and only if acts transitively on .
In particular, for all it follows that and so there exists some such that . In particular is transitive on . Conversely we suppose that is transitive on . Hence for all there exist some such that .
We prove that acts transitively on , so we let and we choose such that . Then, using the transitivity argument above, we let be such that . Then and , which shows the transitive action of on .
Now the definition of an orbital graph implies that is a complete bipartite digraph and hence futile, by Theorem 3.2. ∎
We finish by giving some concrete bounds on the number of edges in futile and non-futile orbital graphs.
Suppose that Hypothesis 2.3 holds. Let , , and be the set of isolated vertices of .
Then is futile if one of the following hold.
and has strictly more than arcs.
and has strictly more than or arcs.
To prove (i) suppose that are distinct and such that . Let be the number of arcs starting in . Now and are not in , so it follows that . We recall that is transitive on , and hence all connected components of have size at least , by Corollary 2.7.
In particular is contained in a connected component of of size at least two, so we deduce from Lemma 2.4 (iii) and (iv) and Lemma 2.9 that for every vertex of , the number of arcs starting there is . Consequently . This means, conversely, that is a complete digraph on as soon as it has strictly more than arcs.
To show (ii) suppose that there are and such that . Let be the number of arcs starting in . As all arcs starting in end in a vertex of it follows that . Let . Then it follows from Lemma 2.4 (iii) that the number of arcs starting in is .
By counting the number of arcs ending in some vertex we obtain, in a similar way, that as well. Hence if has strictly more than or arcs, then is a complete bipartite digraph. ∎
Suppose that Hypothesis 2.3 holds. Then is futile if and only if one of the following conditions is true:
, and .
3.1 Transitive Groups
For transitive groups, it is simpler to identify the futile orbital graphs. These were the first groups where we discovered that some orbital graphs are more useful.
Suppose that Hypothesis 2.3 holds and that acts transitively on .
If acts 2-transitively on , then is futile.
If is futile, then acts 2-transitively on (and hence all orbital graphs are futile).
For (2) we suppose that is futile and we deduce, again by Theorem 3.2, that is a complete digraph or a complete bipartite digraph. The second case is impossible because is transitive on . So is a complete digraph and for any two distinct elements , we deduce that . Then by definition of an orbital graph, there is such that . Hence acts 2-transitively on and the last statement follows from (1). ∎
So we see that for transitive groups if one orbital graph is futile, then all of them are. Lemma 3.7 lets us quickly detect this, as the level of transitivity of a group can be efficiently calculated.
In general these bounds from the lemmas cannot be improved, as the following example shows.
4 Concluding remarks
As mentioned earlier, this work on orbital graphs is motivated by applications in search algorithms for permutation groups. A systematic approach is needed for many open problems and potential applications, and orbital graphs are an interesting class of graphs in their own right. Therefore we phrase some questions, with only some of them being directly related to applications.
Instead of just separating the futile graphs from the useful ones for our algorithms, is it possible to create a finer distinction?
Higman’s Theorem (see for example p.68 in DM ) says that for transitive groups, primitivity can be detected from the orbital graphs.
For imprimitive groups, being able to detect blocks quickly and bring them into a “usefulness analysis” of the graph would be beneficial for computational questions (see Exercise 3.2.14 in DM ).
The theory of association schemes seems to be closely related to orbital graphs. What applications does this have in computational algebra and how do our results relate to this?
There is work in progress on most of these questions.
All authors thank the DFG (Wa 3089/6-1) and the EPSRC CCP CoDiMa (EP/M022641/1) for supporting this work. The second author would like to thank the Royal Society, and the EPSRC (EP/M003728/1). The third author would like to acknowledge support from the OpenDreamKit Horizon 2020 European Research Infrastructures Project (#676541).
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