Ends and vertices of small degree in infinite minimally -(edge)-connected graphs
Bounds on the minimum degree and on the number of vertices attaining it have been much studied for finite edge-/vertex-minimally -connected/-edge-connected graphs. We give an overview of the results known for finite graphs, and show that most of these carry over to infinite graphs if we consider ends of small degree as well as vertices.
1.1 The situation in finite graphs
Four notions of minimality will be of interest in this paper. For , call a graph edge-minimally -connected, resp. edge-minimally -edge-connected if is -connected resp. -edge-connected, but is not, for every edge . Analogously, call vertex-minimally -connected, resp. vertex-minimally -edge-connected if is -connected resp. -edge-connected, but is not, for every vertex . These four classes of graphs often appear in the literature under the names of -minimal/-edge-minimal/-critical/-edge-critical graphs.
It is known that finite graphs which belong to one of the classes defined above have vertices of small degree. In fact, in three of the four cases the trivial lower bound of on the minimum degree is attained. We summarise the known results in the following theorem (some of these results, and similar ones for digraphs, also appear in [1, 9]):
Let be a finite graph, let .
Note that in Theorem 1 (b), the bound of on the degree is best possible. For even , this can be seen by replacing each vertex of , a circle of some length , with a copy of , the complete graph on vertices, and adding all edges between two copies of when the corresponding vertices of are adjacent. This procedure is sometimes called the strong product222The strong product of two graphs and is defined in  as the graph on which has an edge whenever for or , and at the same time either or . of and . For odd values of similar examples can be constructed, using ’s instead of ’s, and in the end deleting two vertices which belong to two adjacent copies of .
In all four cases of Theorem 1, the minimum degree is attained by more than one vertex333We remark that for uniformity of the results to follow, we do not consider the trivial graph to be -edge-connected/-connected.. For convenience let denote the set of all vertices of a graph that have degree at most .
Let be a finite graph, let .
In case (a), actually more than the number of vertices of small degree is known: If we delete all the vertices of degree , we are left with a forest. This was shown in , see also . For extensions of this fact to infinite graphs, see .
The difference in the case in (a) and (c) is due to the paths. For there is no constant in (c): to see this, take the square444The square of a graph is obtained by adding an edge between any two vertices of distance . of any long enough path and add the edge , and the edge . Deleting and we obtain an edge-minimally -edge-connected graph with only six vertices of degree .
The constant from (a) can be chosen as , and this is best possible . Actually one can ensure  that , where denotes the maximum degree of . In (c), the constant may be chosen as about as well (for estimates, see [2, 5, 25]).
The bounds on the number of vertices of small degree are best possible in (b) and (d), for555And for we have (see  for a reference), and this is best possible, as the so-called ladder graphs show. As for , it is easy to see that there are no vertex-minimally -(edge)-connected graphs (since we excluded ). . Indeed, for consider the following example. Take any finite number of copies of the complete graph , and join every two consecutive with a matching of size , in a way that all these matchings are disjoint. Join a new vertex to all vertices of that still have degree , and analogously join a new vertex to half of the vertices of . Finally join and with an edge. See Figure 1.
The obtained graph is vertex-minimally -connected as well as vertex-minimally -edge-connected. However, all vertices but and have degree , which, as , is greater than .
1.2 What happens in infinite graphs?
For infinite graphs, a positive result for case (a) of Theorem 1 has been obtained by Halin  who showed that every infinite locally finite edge-minimally -connected graph has infinitely many vertices of degree , provided that . Mader  extended the result showing that for , every infinite edge-minimally -connected graph has in fact vertices of degree (see Theorem 3 (a) below). It is clear that for , we are dealing with trees, which, if infinite, need not have any vertices of degree .
For the other three cases of Theorem 1, the infinite version fails. In fact, for case (b) this can be seen by considering the strong product of the double-ray (i.e. the two-way infinite path) with the complete graph (cf. Figure 2). The obtained graph is -regular, and vertex-minimally -connected. If instead of the double-ray we take the -regular infinite tree , for any , the degrees of the vertices become unbounded in . For case (d) of Theorem 1 consider the Cartesian product666The Cartesian product of two graphs and is defined [8, 16] as the graph on which has an edge if for or we have that and . of with (see Figure 3).
Counterexamples for an infinite version of (c) will be given now. For the values and this is particularly easy, as for we may consider the double ray , and for its square . All the vertices of these graphs have degree resp. , but and are edge-minimally - resp. -edge-connected.
For arbitrary values , we construct a counterexample as follows. Choose and take the -regular tree . For each vertex in , insert edges between the neigbourhood of in the next level so that spans disjoint copies of (Figure 4 illustrates the case , ). This procedure gives an edge-minimally -edge-connected graph, as one easily verifies. However, the vertices of this graph all have degree at least .
Hence a literal extension of Theorems 1 and 2 to infinite graphs is not true, except for part (a). The reason can be seen most clearly comparing Figures 1 and 2: Where in a finite graph we may force vertices of small degree just because the graph has to end somewhere, in an infinite graph we can ‘escape to infinity’. So an adequate extension of Theorem 1 should also measure something like ‘the degree at infinity’.
This rather vague-sounding statement can be made precise. In fact, the points ‘at infinity’ are nothing but the ends of graphs, a concept which has been introduced by Freudenthal  and later independently by Halin , and which is a mainstay of contemporary infinite graph theory. Ends are defined as equivalence classes of rays (one-way infinite paths), where two rays are equivalent if no finite set of vertices separates them. That this is in fact an equivalence relation is easy to check. The set of all ends of a graph is denoted by . For more on ends consult the infinite chapter of , see also .
The concept of the end degree has been introduced in  and , see also . In fact we distiguish between two types of end degrees: the vertex-degree and the edge-degree. The vertex-degree of an end is defined as the supremum of the cardinalities of the sets of (vertex)-disjoint rays in , and the edge-degree of an end is defined as the supremum of the cardinalities of the sets of edge-disjoint rays in . These suprema are indeed maxima [4, 12]. Note that .
In light of this definition, we observe at once what happens in the case of the infinite version of Theorem 1 (a) above. Edge-minimally -connected graphs, otherwise known as infinite trees, need not have vertices that are leaves, but if not, then they must have ‘leaf-like’ ends, that is, ends of vertex-degree . In fact, it is easy to see that in a tree , with root , say, every ray starting at corresponds to an end of , and that all ends of have vertex- and edge-degree . On the other hand, rayless trees have leaves.
This observation gives case (a’) in the following generalisation of Theorem 1 to infinite graphs. Cases (b)–(d) of Theorem 3, respectively their quantative versions in Theorem 4, are the main result of this paper.
Let be a graph, let .
(Mader ) If is edge-minimally -connected and , then has a vertex of degree .
If is edge-minimally -connected, then has a vertex of degree or an end of edge-degree .
If is vertex-minimally -connected, then has a vertex of degree or an end of vertex-degree .
If is edge-minimally -edge-connected, then has a vertex of degree or an end of edge-degree .
If is vertex-minimally -edge-connected, then has a vertex of degree or an end of vertex-degree .
As in the finite case, one can give bounds on the number of vertices/ends of small degree. Recall that denotes the set of all vertices of degree at most , and let resp. denote the set of ends of vertex- resp. edge-degree at most .
Concerning part (c) we remark that we may replace graphs with multigraphs (see Corollary 12). Also, in (a’) and (c), one may replace the edge-degree with the vertex-degree, as this yields a weaker statement.
We shall prove Theorem 4 (b), (c) and (d) in Sections 2, 3 and 4 respectively. Statement (a’) is fairly simple, in fact, it follows from our remark above that every tree has at least two leaves/ends of vertex-degree . In general, this is already the best bound, because of the (finite or infinite) paths. For trees of uncountable order we get more, as these have to contain a vertex of degree , and it is then easy to find vertices/ends of (edge)-degree .
In analogy to the finite case, the bounds on the degrees of the vertices in (b) cannot be lowered, even if we allow the ends to have larger vertex-degree. An example for this is given at the end of Section 2. There, we also state a lemma that says that the vertex-/edge-degree of the ends in Theorem 4 will in general not be less than .
Also, the bound on the number of vertices/ends of small degree in Theorem 4 (b) and (d) is best possible. For (d), this can be seen by considering the Cartesian product of the double ray with the complete graph (for that is the double-ladder). For (b), we may again consider the strong product of the double ray with the complete graph (see Figure 2 for ). The latter example also shows that in (b), we cannot replace the vertex-degree with the edge-degree.
As for Theorem 4 (c), it might be possible that the bound of Theorem 2 (c) extends. For infinite graphs , the positive proportion of the vertices there should translate to an infinite set of vertices and ends of small degree/edge-degree. More precisely, one would wish for a set of cardinality , or even stronger, . Observe that it is necessary to exclude also in the infinite case the two exceptional values and , as there are graphs (e.g. and ) with only two vertices/ends of (edge)-degree resp. .
For , does every infinite edge-minimally -edge-connected graph contain infinitely many vertices or ends of (edge)-degree ? Does have , or even , such vertices or ends?
Another interesting question is which -(edge)-connected graphs have vertex- or edge-minimally -(edge)-connected subgraphs. Especially interesting in the case of edge-connectivity would be an edge-minimally -edge-connected subgraph on the same vertex set as the original graph. Finite graphs trivially do have such subgraphs, but for infinite graphs this is not always true. One example in this respect is the double-ladder, which is -connected but has no edge-minimally -connected subgraphs on the same vertex set. This observation leads to the study of vertex-/edge-minimally -(edge)-connected (standard) subspaces rather than graphs. For more on this, see [7, 26], the latter of which contains a version of Theorem 3 (a) for standard subspaces.
We finish the introduction with a few necessary definitions. The vertex-boundary of a subgraph of a graph is the set of all vertices of that have neighbours in . The edge-boundary of is the set . A region of a graph is a connected induced subgraph with finite vertex-boundary . If , then we call a -region of . A profound region is a region with .
2 Vertex-minimally -connected graphs
In this section we shall show part (b) of Theorem 4. For the proof, we need two lemmas. The first of these lemmas may be extracted777Although the graphs there are all finite, the procedure is the same. from  or from , and at once implies Theorem 1 (b). For completeness, we shall give a proof.
Let , let be a vertex-minimally -connected graph, and let be a profound finite -region of . Then has a vertex of degree at most .
Moreover, if , then we may choose .
Note that we may assume is inclusion-minimal with the above properties. Set , set , and set . Let , and observe that since is vertex-minimally -connected, there is a -separator of with . Let be a component of , set , and set . Furthermore, for set and set . Observe that .
We claim that there are with either or such that for :
In fact, observe that for we have that Thus either , which by the minimality of implies that is empty, or , which by the -connectivity of implies that is empty. This proves (1).
We hence know that there is an such that . Now, as , (1) implies that
Thus, there is a vertex of degree at most
We also need a lemma from .
[27, Lemma 5.2] Let be a graph such that all its ends have vertex-degree at least . Let be an infinite region of . Then there exists a profound region for which one of the following holds:
is finite and , or
is infinite and for every profound region .
Observe that the outcome of Lemma 7 is invariant under modifications of the structure of . Hence in all applications we may assume that only for ends of that have rays in .
We are now ready to prove Theorem 4 (b).
Proof of Theorem 4 (b).
First of all, we claim that for every infinite region of it holds that:
|There is a vertex of degree or an end of vertex-degree with rays in .||(3)|
In order to see (3), we assume that there is no end as desired and apply Lemma 7 to with . This yields a profound region . We claim that (a) of Lemma 7 holds; then we may use Lemma 6 to find a vertex with .
So, assume for contradiction that (b) of Lemma 7 holds. Since is -connected there exists a finite family of finite paths in such that each pair of vertices from is connected by pairwise internally disjoint paths from . Set
and observe that is still infinite. In particular, contains a vertex .
Since is vertex-minimally -connected, lies in a -separator of . By the choice of , no two vertices of are separated by . Thus all of is contained in one component of .
Let be a component of that does not contain any vertices from . Note that as is -connected, has a neighbour in . Hence , and is a profound region with .
In fact, , which is clear if , and otherwise follows from the fact that and thus, because we know that . So, (b) implies that , a contradiction as desired. This proves (3).888Observe that taking , we have thus proved Theorem 3 (b).
Now, let be any separator of of size (a such exists by the vertex-minimality of ). First suppose that has at least one infinite component . Then we apply Lemma 6 or (3) to any component of and find an end of vertex-degree with no rays in , or a vertex of degree at most . Let denote the point found, that is, or .
Let be the subgraph of induced by and all vertices of that have infinite degree into . Then is a region, and we may thus apply (3) to in order to find the second end/vertex of small (vertex)-degree. This second point is different from by the choice of .
It remains to treat the case when all components of are finite. As we otherwise apply Theorem 2 (b), we may assume that has infinitely many components. Hence, as has no -separators, each has infinite degree. This means that we may apply Lemma 6 to any two -regions , with in order to find two vertices , , each of degree . ∎
We remark that the bound on the vertex-degree given by Theorems 3 (b) and 4 (b) is best possible. Indeed, by the following lemma, which follows from Lemma 7.1 from , the vertex-degree of the ends of a -connected locally finite graph has to be at least .
Let , let be a locally finite graph, and let . Then if and only if is the smallest integer such that every finite set can be separated999We say a set separates a set from an end if the unique component of that contains rays of does not contain vertices from . from with a set of vertices.
For non-locally finite graphs, the minimum size of an – separator corresponds to the vertex-/edge-degree of plus the number of dominating vertices of . See .
One might now ask whether it is possible to to achieve a better upper bound on the degree of the ‘small degree vertices’ than the one given by Theorems 3 and 4 (b), if one accepts a worse bound on the vertex-degree of the ‘small degree ends’. The answer is no. This is illustrated by the following example for even (and for odd there are similar examples).
Let , and take the disjoint union of double-rays . For simplicity, assume that divides . For each , take copies of the strong product of with , and identify the vertices that belong to the first or the last copy of with the th vertices the . This can be done in a way so that the obtained graph, which is easily seen to be vertex-minimally -connected, has two ends of vertex-degree , while the vertices have degree either or .
3 Edge-minimally -edge-connected graphs
Let be a graph and let be a sequence of regions such that for all . Then there is an end that has a ray in each of the so that
if for all , then , and
if for all , then .
As all the are connected, it is easy to construct a ray which has a subray in each of the . Say belongs to the end . We only show (i), as (ii) can be proved analogously.
Suppose for contradiction that for all , but . Then contains a set of disjoint rays. Let be the set of all starting vertices of these rays. Since for all , there is an such that . (To be precise, one may take .) But then, it is impossible that all rays of have subrays in , as only of them can pass disjointly through . ∎
We also need the following lemma from .
[27, Lemma 3.2] Let and let be a region of a graph so that and so that for every non-empty region of . Then there is an inclusion-minimal non-empty region with .
Combined, the two lemmas yield a lemma similar to Lemma 7 from the previous section:
Let be a region of a graph so that and so that for every end with rays in . Then there is an inclusion-minimal non-empty region with .
Set and inductively for , choose a non-empty region such that (if such a region exists). If at some step we are unable to find a region as above, then we apply Lemma 10 to to find the desired region . On the other hand, if we end up defining an infinite sequence of regions, then Lemma 9 (ii) tells us that there is an end with rays in and , a contradiction. ∎
We are now ready to prove part (c) of our main theorem:
Proof of Theorem 4 (c).
Since is edge-minimally -edge-connected, has a non-empty region such that , and such that . We shall find a vertex or end of small (edge)-degree in ; then one may repeat the procedure for in order to find the second point.
First, we apply Lemma 11 with to obtain an end as desired or an inclusion-minimal non-empty region with . If should consist of only one vertex, then this vertex has degree , as desired. So suppose that has more than one vertex, that is, is not empty.
Let . By the edge-minimal -edge-connectivity of we know that belongs to some cut of with . Say where partition . Since , neither nor is empty.
So, and , by the minimality of . But then, since and , we obtain that
Hence, at least one of , , say the former, is strictly smaller than . Since is -edge-connected, this implies that is empty. But then , a contradiction to the minimality of . ∎
We dedicate the rest of this section to multigraphs, that is, graphs with parallel edges, which often appear to be the more appropriate objects when studying edge-connectivity. Defining the edge-degree of an end of a multigraph in the same way as for graphs, that is, as the supremum of the cardinalities of the sets of edge-disjoint rays from , and defining and as earlier for graphs, we may apply the proof of Theorem 4 (c) with only small modifications101010We will then have to use a version of Lemma 10 for multigraphs. Observe that such a version holds, as we may apply Lemma 10 to the (simple) graph obtained by subdividing all edges of the multigraph. This procedure will not affect the degrees of the ends. The rest of the proof will then go through replacing everywhere ‘graph’ with ‘multigraph’. to multigraphs. We thus get:
Let be an edge-minimally -edge-connected multigraph. Then .
In particular, every finite edge-minimally -edge-connected multigraph has at least two vertices of degree .
However, a statement in the spirit of Theorem 2 (c) does not hold for multigraphs, no matter whether they are finite or not. For this, it suffices to consider the graph we obtain by multiplying the edges of a finite or infinite path by . This operation results in an edge-minimally -edge-connected multigraph which has no more than the two vertices/ends of (edge)-degree which were promised by Corollary 12.
4 Vertex-minimally -edge-connected graphs
In this section we prove Theorem 4 (d). The proof is based on Lemma 14, which at once yields Theorem 2 (d), the finite version of Theorem 4 (d). The idea of the proof of this lemma (in particular Lemma 13) is similar to Mader’s original proof of Theorem 2 (d) in .111111But as  does not contain the statement we need for finding the second vertex/end of small degree, we cannot make use of it here.
We need one auxiliary lemma before we get to Lemma 14. For a set in a graph write and .
Let . Let be a graph, let with , and let be a component of so that . Then contains a vertex of degree at most .
Suppose that the vertices of all have degree at least . Then each sends at least edges to . This means that
So , which, as , is only possible if both sides of the inequality are negative, that is, if . But this is impossible, as . ∎
As usal, the edge-connectivity of a graph is denoted by . Also, in order to make clear which underlying graph we are referring to, it will be useful to write for a region of a graph .
Let , let be a -edge-connected graph, and let be an inclusion-minimal region of with the property that has a vertex so that and . Suppose for each , the graph has a cut of size . Then contains a vertex of degree (in ).
If every vertex of has a neighbour in then we may apply Lemma 13 with and are done. So let us assume that there is a vertex all of whose neighbours lie in . By assumption, has a cut of size , which splits into and , with , say. See Figure 6.
Since is -edge-connected, is not a cut of . Hence has neighbours in both and . Thus, as , and , it follows that . Consider the region induced by and . Because , we know that .
So, by the choice of and we may assume that . Thus,
implying that . That is, (here we use again that ). As , this contradicts the minimality of . ∎
Any finite vertex-minimally -edge-connected graph clearly has an inclusion-minimal region as in Lemma 14. Thus Theorem 1 (d) follows at once from Lemma 14. Applying Lemma 14 to any inclusion-minimal region with the desired properties that is contained in in order to find a second vertex of small degree in , we get:
Corollary 15 (Theorem 2 (d)).
Let be a finite vertex-minimally -edge-connected graph. Then has at least two vertices of degree .
This means that for a proof of Theorem 4 (d) we only need to worry about the infinite regions, which is accomplished in the next lemma.
Let , let be a vertex-minimally -edge-connected graph and let be a region of . Let such that . Suppose has no inclusion-minimal region with the property that contains a vertex so that . Then has an end of vertex-degree with rays in .
We construct a sequence of infinite regions of , starting with which clearly is infinite. Our regions will have the property that , which means that we may apply Lemma 9 (i) in order to find an end as desired.
In step , for each pair of vertices in , take a set of edge-disjoint paths joining them: the union of all these paths gives a finite subgraph of . Since was infinite, still is, and thus contains a vertex .
Since is vertex-minimally -edge-connected, has a cut of size less than , which splits into and , say, which we may assume to be connected. Say contains a vertex of . Then (since ). Thus, as has neighbours in both and (because is -connected), we obtain that . Observe that is infinite, as otherwise it would contain an inclusion-minimal region as prohibited in the statement of the lemma. ∎
We finally prove Theorem 4 (d).
Proof of Theorem 4 (d).
Let , and let be a cut of with . Say splits into and . For , if contains an inclusion-minimal region such that has a vertex with the property that , we use Lemma 14 to find a vertex of degree at most in . On the other hand, if does not contain such a region, we use Lemma 16 to find an end of the desired vertex-degree. ∎
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