ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 17 (2019) 581-589
https://doi.org/10.26493/1855-3974.1866.fd9
(Also available at http://amc-journal.eu)
Lobe, edge, and arc transitivity of graphs of connectivity 1
Jack E. Graver, Mark E. Watkins
Department of Mathematics, Syracuse University, Syracuse, NY
Received 27 November 2018, accepted 3 October 2019, published online 9 December 2019
We give necessary and sufficient conditions for lobe-transitivity of locally finite and locally countable graphs whose connectivity equals 1. We show further that, given any biconnected graph A and a "code" assigned to each orbit of Aut(A), there exists a unique lobe-transitive graph r of connectivity 1 whose lobes are copies of A and is consistent with the given code at every vertex of r. These results lead to necessary and sufficient conditions for a graph of connectivity 1 to be edge-transitive and to be arc-transitive. Countable graphs of connectivity 1 the action of whose automorphism groups is, respectively, vertex-transitive, primitive, regular, Cayley, and Frobenius had been previously characterized in the literature.
Keywords: Lobe, lobe-transitive, edge-transitive, orbit, connectivity. Math. Subj. Class.: 05C25, 05C63, 05C38, 20B27
1 Introduction
Throughout this article, r denotes a connected simple graph. Consider the equivalence relation = on the edge-set £T of r whereby e1 = e2 whenever the edges e1 and e2 lie on a common cycle of r. A lobe is a subgraph of r induced by an equivalence class with respect to =. Equivalently, a lobe is a subgraph that either consists of a cut-edge with its two incident vertices or is a maximal biconnected subgraph1. A vertex is a cut-vertex if it belongs to at least two different lobes. Connected graphs other than K2 have connectivity 1 if and only if they have a cut-vertex. Clearly no finite vertex-transitive graph admits a cut-vertex.
E-mail addresses: jegraver@syr.edu (JackE. Graver), mewatkin@syr.edu (MarkE. Watkins)
1The term "lobe" is due to O. Ore [6]. We eschew the term "block" for this purpose, as it leads to ambiguity when discussing imprimitivity.
Abstract
©® This work is licensed under https://creativecommons.Org/licenses/by/4.0/
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Graphs of connectivity 1 whose automorphism groups have certain given properties have been characterized. Those whose automorphism groups are, respectively, vertex-transitive, primitive, and regular were characterized in [5]. In particular, primitive planar graphs of connectivity 1 were characterized in [11]2. Cayley graphs of connectivity 1 were characterized in [9]. Graphs of connectivity 1 with Frobenius automorphism groups were characterized in [10]. In the present work, we complete this investigation; we characterize graphs of connectivity 1 whose automorphism groups act transitively on their set of lobes. As a consequence, we obtain characterizations of edge-transitive graphs and arc-transitive graphs of connectivity 1.
The conditions for a graph of connectivity 1 to be lobe-transitive or to be vertex-transitive are independent; such a graph may have either property or neither one or both. Such is not the case for edge- and arc-transitivity. In Section 3 we give necessary and sufficient conditions for a graph to be lobe-transitive. We further show that, given any biconnected graph A and a "list" of orbit-multiplicities of copies of Aut(A), one can construct a lobe-transitive graph of connectivity 1 all of whose lobes are isomorphic to A and locally respects the given list. We give necessary and sufficient conditions for a countable graph of connectivity 1 to be edge-transitive in Section 4 and to be arc-transitive in Section 5. As the sets of conditions for these latter two properties are more intertwined with lobe-transitivity than the characterization of vertex-transitivity (for graphs of connectivity 1), scattered throughout are examples that illustrate some algebraic distinctions among these various properties.
2 Preliminaries
Throughout this article, the symbol N denotes the set of positive integers. The symbols I, J, and K, often subscripted, denote subsets of N of the form {1,2,... ,n} or the set N itself; they appear as sets of indices. All graphs (and their valences) in this article are finite or countably infinite. The symbol Si,j (the so-called "Kronecker delta") assumes the value 1 if i = j and 0 if i = j. For a graph A and any subgroup H < Aut(A), the set of orbits of H acting on VA is denoted by O(H).
The set of lobes of a graph r is denoted by L(r). We let {Lk : k e K} denote the partition of L(r) into isomorphism classes of lobes. For given k e K and a lobe A e Lk, we let O (Aut(A)) = {(VAj : j e Jk}, and we understand that if a: A ^ © is an isomorphism between lobes in Lk, then a((VA)j) = (V©)j for all j e Jk. Finally, for each k e K and j e Jk, we define the function r(k): Vr ^ N by
Tj(k)(v) = |{A e Lk : v e (VAj}|.	(2.1)
For A0 e L(r) and n e N, we recursively define the subgraphs
ro(Ao) = Ao, r„+i(Ao) = U{A e L(r) : VA n Vr„(Ao) = 0}.
Lemma 2.1 ([5, Lemma 3.1]). Let A, © G L (T) and let n G N. If for each k G K and j G Jk, the function rjfc) is constant on Vr, then any isomorphism an: rn (A) ^ rn(©) admits an extension to an automorphism a G Aut(r).
2For a short algebraic proof that all 1-ended planar graphs with primitive automorphism group are biconnected, see [8].
J. E. Graver and M. E. Watkins: Lobe, edge, and arc transitivity of graphs of connectivity 1 583
This lemma was used in [5] to prove the following characterization of vertex-transitive graphs of connectivity 1.
Theorem 2.2 ([5, Theorem 3.2]). Let r be a graph of connectivity 1. A necessary and suffic
V r.
(k)
sufficient condition for r to be vertex-transitive is that all the functions t( ) be constant on
Notation. When all the lobes of the graph r are pairwise isomorphic, that is, the index set K has but one element, then in Equation (2.1) the index k is suppressed; we simply replace
Jfc by J and j by Tj.
3 Lobe-transitivity
Let r be a graph of connectivity 1. It is immediate from the above definitions that the edge-sets of the lobes of r are blocks of imprimitivity of the group Aut(r) acting on Er. Hence any automorphism of r must map lobes onto lobes, and therefore, if Aut(r) is to act transitively on L(r), then all the lobes of r must be pairwise isomorphic. However, pairwise-isomorphism of the lobes alone is not sufficient for lobe-transitivity, even when every vertex of r lies in the same number of lobes.
Let us first dispense with trees; the proof is elementary and hence omitted.
Proposition 3.1. A finite or countable tree is lobe-transitive (and simultaneously, edge-transitive) if and only if there exist nl,n2 G N U{K0} such that every edge has one incident vertex of valence n and the other of valence n2. If n = n2, the tree is also arc-transitive.
For graphs of connectivity 1 other than trees, we have the following characterization of lobe-transitivity.
Theorem 3.2. Let r be a graph of connectivity 1, and let A0 be an arbitrary lobe of r. Let {Pi : i G I} be the set of orbits of Aut(r), and let Q = {Qj : j G J} be the set of those orbits of the stabilizer in Aut(r) of A0 that are contained in A0. Then necessary and sufficient conditions for the graph r to be lobe-transitive are:
(1)	For each lobe A G L (r), there exists an isomorphism aA : A0 ^ A.
(2)	For each j G J, there exists a function Tj : Vr ^ N U {0, K0} such that
(a)	for all v G Vr,
Tj(v) = |{A G L(r): v G aA(Qj)}|	(3.1)
and
(b)	for each i G I, Tj is constant on Pi and is nonzero if and only if Qj C Pi.
Proof. (Necessity) Suppose that r is lobe-transitive. For each lobe A G L(r), there is an automorphism aA G Aut(r) that maps the fixed lobe A0 onto A. The restriction to A0 of aA is an isomorphism aA: A0 ^ A that satisfies condition (1).
For any lobe A, an automorphism a G Aut(r) is in the stabilizer of A if and only if a—1aaA is in the stabilizer of A0. It follows that the partition {aA(Qj) : j G J} of VA is the set of orbits of the stabilizer of A that are contained in A. Furthermore, since the stabilizer of A0 is a subgroup of Aut(r), the partition {aA(Qj) : j G J} of VA
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refines the partition {Pj n VA : i e I}. If for some indices i and j, the vertex v satisfies v e aA(Qj) C Pj, then for any lobe ©, the vertex a0a-1(v) lies in Pj n ae(Qj). This implies that, for all j e J, the function Tj as given in Equation (3.1) is well-defined and constant on Pj.
Suppose that for an arbitrary index i e I, the vertex v lies in Pj. Since by Equation (3.1), Tj (v) counts for each j e J the number of lobes A such that v lies in aA(Qj ), it follows that Tj (v) is positive exactly when a-1(v) e Qj C Pj holds, concluding the proof of condition (2.b).
(Sufficiency) Assume conditions (1) and (2). To prove that r is lobe-transitive, it suffices to prove that every isomorphism ae : A0 ^ © is extendable to an automorphism of r. (Note that in this direction of the proof, ae is not presumed to be the restriction to A0 of an automorphism ae e Aut(r) but, in fact, it is.)
Fix a lobe ©0 and a vertex v e VA0 and let w = a0o (v). For some j e J, the vertex v lies in Qj, and so w e aeo (Qj). Since both Tj(v) and Tj(w) are therefore positive, both v and w lie in the same orbit Pj of Aut(r) by condition (2.b). Furthermore, since Tj is constant on Pj for each j e J, there exists a bijection Pj from the set of lobes A such that v e aA(Qj) onto the set of lobes © such that w e ae(Qj). Let A1 be a lobe in the former set, and let ©1 = Pj(A1).
Although v and w lie in the images of the same orbit Qj in lobes A1 and ©1, respectively, the vertices a-^v) and ct—1(w) need not be the same vertex of A0. However, since both vertices lie in the same orbit Qj of A0, there exists an automorphism a e Aut(A0) such that aa-1(v) = a-1(w). Then aei aa— is an isomorphism from A1 onto ©1 that maps v onto w and therefore agrees with aeo at the vertex v common to A0 and A1.
The amalgamation of ct01 aa-1 with aeo is an isomorphism from A0 U A1 to ©0 U ©1. By repeating this same technique, we can extend aeo to all lobes adjacent to A0 and then to all of their adjacent lobes and inductively to all of r.	□
Example 3.3. Suppose that the lobes of r are copies of some biconnected, vertex-transitive graph and that every vertex of r is incident with exactly m lobes where m > 2. By Theorem 2.2, r is vertex-transitive. By Theorem 3.2, r is lobe-transitive, with O(Aut(r)) being the trivial partition (with just one big cell P1). Also |J| = 1 and t1(v) = m for all
v e vr.
Remark 3.4. There exists a "degenerate" family of lobe-transitive graphs r of connectivity 1 that have but a single cut-vertex. For some cardinal K > 2, consider a collection of K copies of a biconnected graph A0, and let v0 e VA0. Let aA : A0 ^ A be an isomorphism as in Theorem 3.2, and let aA(v0) = vA for each copy A of A0 in the collection. We obtain r by identifying v0 and all the vertices vA and naming the new amalgamated vertex w, which forms a singleton orbit {w} of Aut(r). Clearly r is lobe-transitive and w is its unique cut-vertex. If A0 has finite diameter, then r has zero ends (see [3]) when A0 is finite and has K ends when A0 is infinite; if A0 has infinite diameter, then r has at least K ends. Other than the graphs just described, all countable lobe-transitive graphs of connectivity 1 are "tree-like" with H0 cut vertices and either 2 or 2N° ends.
Theorem 3.5. Let A0 be any biconnected graph. Let H < Aut(A0), let Q = O(H) = {Qj : j e J}, and let R = {Rk : k e K} be a partition of VA0 refined by Q. For each k e K, let the function : J ^ N U {0, H0} satisfy (j) > 0 if and only if Qj C and (to avoid the triviality of a single lobe) jej Mfc(j) > 2 for at least one k e K. Then there exists (up to isomorphism) a unique lobe-transitive graph r of connectivity 1 such that
J. E. Graver and M. E. Watkins: Lobe, edge, and arc transitivity of graphs of connectivity 1 585
(1)	for each lobe A G L (T), there exists an isomorphism aA : Ao ^ A;
(2)	for each vertex v G Vr and each j G J, we have
Mfc(j) = |{A G L(r) : a_^v) G Qj C Rfc}|.
Proof. Let A0, H, Q, and be as postulated. Let r0 = A0 from which we construct r as follows.
Let v be any vertex of A0. For some j, k, it must hold that v G Qj C Rk, and so Mfc(j) > 0. For each I such that Qe C Rk, we postulate the existence of (¿) copies A of A0 (including A0 itself when I = j) such that, if aA: A0 ^ A is an isomorphism, then some vertex in aA(Qe) is identified with the vertex v. The graph r is produced by repeating this process for each vertex of A0. We repeat this process starting at each vertex w G Vr \ Vr0, the only notational change being that, if specifically w G aA(Qj/) for some j' G J, then we consider the subset aA(Qj) of VA (instead of Qj in A0) to which w belongs. Thus we construct r2.
Inductively, suppose that rn has been constructed for some n > 2. Let w G Vrn \ Vrn_i, and so w G aA(Qm) holds for some m G J and a unique lobe A G L(rn) \ L(rn-1). Supposing that Qm C Rk, we postulate the existence of (m) new copies of A0 that share only the vertex w with rn according to the above identification. In this way we construct rn+1. Finally, let r = |Jrn.
It remains only to prove that r so-constructed is lobe-transitive. Let © be any lobe of r. By the above construction, all lobes of r are pairwise isomorphic, and so there exists an isomorphism a0 : A0 ^ ©. Starting with r0 = © and by using the technique in the proof of Sufficiency in Theorem 3.2, one constructs a sequence r0, ri,... so that for all n G N, we have = rn, and a0 is extendable to an isomorphism from rn to r^. Thus r — Urn, and a0 can be extended to an automorphism of r.	□
Example 3.6. In the notation of Theorem 3.5, let A0 be the 5-cycle with one chord as shown in Figure 1(a), and let H = Aut(A), yielding the orbit partition {Q1, Q2, Q3} as indicated. Let Ri = Qi U Q3 and R2 = Q2, giving J = {1,2, 3} and K = {1,2}. Define ^1(1) = 3, ^1(3) = 1, and ^2(2) = 2. Note that all other values of and must equal 0. Then r1 is as seen in Figure 1(b).
Q1 Q2 Q3
(a) Aq
(b) ri
Figure 1: A0 and r1 from Example 3.6.
The pairs of conditions in Theorems 3.2 and 3.5 may appear alike, but there is a notable difference between them. This occurs when the arbitrarily chosen subgroup H < Aut(A0)
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of Theorem 3.5 is a proper subgroup of the stabilizer of A0 in Aut(T), where r is the graph constructed from A0 and the functions of Theorem 3.5. We illustrate this distinction with following example.
Example 3.7. Our initial lobe A0 is a copy of K4, with vertices labeled as in Figure 2(a), and so Aut(A0) = Sym(4) of order 24. We use A0 to "build" the lobe-transitive graph r shown four times in Figure 2(b). The action on A0 by the stabilizer of A0 in Aut(r) is the 4-element group (gi) x (g2) whose generators have cycle representation g1 = (vi, v2) and g2 = (v3, v4). The shadings of the vertices in the four depictions of r in Figure 2(b) correspond respectively to the four different subgroups of (g1) x (g2) described below. For the sake of simplicity, we assume R = Q.
(a)	(b)
Figure 2: The clothesline graph.
(i)	H is the trivial group {i}. Thus H induces four orbits Qj = {vj} for j e {1, 2, 3,4}. The functions are then given by (j) = 2j for k e {1,2} and (j) = Sj,k for k e {3,4}.
(ii)	H = (gi}. There are three orbits of H: Qi = {vi, v2}, Q2 = {v3}, and Q3 = {v4}, which give ^(1) = 2, ^2(2) = ^3(3) = 1. All other functional values are zero.
(iii)	H = (g2}. Again there are three orbits of H but not the same ones: Q1 = {v1}, Q2 = {v2}, and Q3 = {v3,v4}. This gives ^fc(j) = 2Sj,k for k e {1,2} and
M3(j) = Sj,3.
(iv)	H = (g1} x (g2}. Now there are just two orbits: Q1 = {v1, v2} and Q2 = {v3, v4}. Finally, ^1(1) = 2, ^2(2) = 1, and all other functional values are zero.
All four choices for H, the partition Q, and the functions clearly yield the same lobetransitive graph r of connectivity 1 by the construction of Theorem 3.5
4 Edge-transitivity
Lemma 4.1. If r is an edge-transitive (respectively, arc-transitive) graph, then r is lobetransitive and its lobes are also edge-transitive (respectively, arc-transitive).
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Proof. For i = 1,2, let ©i G L(T), and let ei be an edge (respectively, arc) of ©¿. There exists an an automorphism p G Aut(r) such that p(e1) = e2. Since p maps cycles through ei onto cycles through e2, p must map ©1 onto ©2. If e1 and e2 lie in the same lobe ©, then p leaves © invariant, and so its restriction to © is an automorphism of ©.	□
Theorem 4.2. Let r be a graph of connectivity 1 with more than one lobe, and let A G L (r). Necessary and sufficient conditions for r to be edge-transitive are the following:
(1)	The lobes of r are edge-transitive.
(2)	For each lobe © G L(r), there exists an isomorphism a© : A ^ ©.
(3)	Exactly one of the following descriptions of r holds:
(a)	Both r and A are vertex-transitive, in which case every vertex is incident with the same number > 2 of lobes.
(b)	The graph r is vertex-transitive but A is not vertex-transitive, in which case A is bipartite with bipartition {Q1, Q2}, and there exist constants m1,m2 G N U {Ho} such that for j = 1, 2 and all v G V r, it holds that
mj = |{© G L(r): v G a©(Qj)}|.
(c)	The graph r is not vertex-transitive, in which case r is bipartite with bipartition {P1, P2} and there exist constants m1,m2 G N U {H0}, at least one of which is at least 2, such that for i =1,2, if v G Pi, then
|{© G L (r) : v G a © (Pj n V A)}| = mj Sitj.
Proof. (Sufficiency) Assume all the conditions in the hypothesis and let e1, e2 G £T be arbitrary edges in lobes A1 and A2, respectively. By condition (2), there exists an isomorphism a: A1 ^ A2. By condition (1), there exists an automorphism a G Aut(A2) such that e2 = aa(e1). Each of the three cases of condition (3) is seen to satisfy the hypothesis of Lemma 2.1, implying that aa: A1 ^ A2 is extendable to an isomorphism of r mapping e1 to e2 .
(Necessity) Suppose that r is an edge-transitive graph of connectivity 1. By Lemma 4.1, r is also lobe-transitive and its lobes are edge-transitive, proving condition (1). Condition (2), which establishes notation for the remainder of this proof, also follows from Lemma 4.1.
To prove (3), we continue the notation of Theorem 3.2 with I being the index set for the set of orbits of Aut(r) and J being the index set for the orbits of the stabilizer of A that are contained in A. Since both r and all of its lobes are edge-transitive, |I| and |J| equal either 1 or 2.
If both r and A are vertex-transitive, then |I| = | J| = 1, and for every vertex v G Vr, t1 (v) = m holds for some m > 2. This is case (3.a).
Since any odd cycle in r would be contained in a lobe of r, it holds that r is bipartite if and only if every lobe is bipartite. If either r or A is not vertex-transitive, then each -and hence both - are bipartite, and the sides of the bipartitions (whether or not they are entire orbits of the appropriate automorphism group) are blocks of imprimitivity systems. Let {P1, P2} be the bipartition of Vr, and so {P1 n V©, P2 n V©} is the bipartition of
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any lobe ©. Equivalently, letting {Qi, Q2} denote the bipartition of A, we have Pj = UeeL(r) CTe(Qi) for i = 1,2.
Suppose that r is vertex-transitive but A is not, and so |I| = 1 and |J| = 2. By Theorem 2.2, there exist constants m1, m2 G N U {H0} such that for all v G Vr and j G J, we have mj = Tj(v) = |{0 G L(r) : v G ae(Qj)}|.
Finally, suppose that r is not vertex-transitive, and so P1 and P2 are the orbits of Aut(r). Also, A is bipartite with bipartition {Q1, Q2}, where Qj = Pj n VA. As no automorphism of r swaps P1 with P2, no automorphism of r swaps Q1 with Q2 (even though A may be vertex-transitive!). Hence |I| = |J| = 2. Since r is lobe-transitive, it follows now from the "necessity" argument of Theorem 3.2 that, for j = 1, 2, the function Tj satisfies the condition Tj(v) > 0 if and only if v G Pj. That means that there exist constants m1, m2 G N U {K0}, at least one of which is greater than 1, such that, if v G Pj, then Tj (v) = mjSj,j.	□
Example 4.3. Suppose in the notation of Theorem 4.2 that r is edge-transitive and A is the complete bipartite graph Ks,t with |Q1| = s and |Q2| = t. Suppose that every vertex of r is incident with exactly two lobes isomorphic to A. If s = t, then both r and A are vertex-transitive, and we have case (3.a) of the theorem. If s = t and every vertex lies in one image of Q1 and one image of Q2, then we have the situation of case (3.b). If again s = t but each vertex lies in either two images of Q1 or two images of Q2, then we have the situation described in case (3.c).
Remark 4.4. With regard to Example 4.3, we note that having s = t does not assure vertex-transitivity of edge-transitive bipartite graphs. There exist edge-transitive, non-vertex-transitive, finite bipartite graphs where the two sides of the bipartition have the same size. Such graphs are called semisymmetric. The smallest such graph, on 20 vertices with valence 4, was found by J. Folkman [2], who also found several infinite families of semisymmetric graphs. Many more such families as well as forbidden values for s were determined by A. V. Ivanov [4].
Example 4.5. This simple example illustrates how the converse of Lemma 4.1 is false, even though the lobes themselves may be highly symmetric. Let r be a graph of connectivity 1 whose lobes are copies of the Petersen graph (which is 3-arc-transitive!). For each lobe A, let (VA)1 and (VA)2 denote the vertex sets of disjoint 5-cycles indexed "consistently," i.e., if A and © share a vertex v, then v g (VA) j n (V©) j for i =1 or i = 2. (Observe that r is not bipartite.) For i = 1, 2 define Pj = |JeejZ,(r) (V©) j, and suppose that each vertex in Pj belongs to exactly mj lobes. The graph r is lobe-transitive by Theorem 3.2, and r is both vertex- and edge-transitive when m1 = m2, but r is neither vertex- nor edge-transitive when m1 = m2.
5 Arc-transitivity
Theorem 5.1. Let r be a graph of connectivity 1. Necessary and sufficient conditions for r to be arc-transitive are the following:
(1)	The lobes of r are arc-transitive.
(2)	The lobes of r are pairwise isomorphic.
(3)	All vertices of r are incident with the same number of lobes.
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Proof. (Necessity) Suppose that r is arc-transitive. Conditions (1) and (2) follow from Lemma 4.1. Since arc-transitivity implies vertex-transitivity, condition (3) holds.
(Sufficiency) Assume that the three conditions hold. For k =1,2, let ak be an arc of r, and let ©k be the lobe containing ak. By condition (2), there exists an isomorphism a : ©i ^ ©2. By condition (1), there exists an automorphism a e Aut(©2) such that aa(ai) = a2. By condition (3), the functions Tj of Equation (2.1) are constant on Vr. (In fact, since the lobes are vertex-transitive, there is only one such function.) It now follows from Lemma 2.1 that aa is extendable to all of r.	□
Remark 5.2. If conditions (1) and (3) of Theorem 5.1 were replaced by the lobes are edge-transitive and r is vertex-transitive, the sufficiency argument would fail. There exist finite graphs [1] and countably infinite graphs with polynomial growth rate [7] that are vertex-and edge-transitive but not arc-transitive. Let A denote such a graph, and consider a graph r whose lobes are isomorphic to A with the same number of lobes incident with every vertex. Then r itself is vertex- and edge-transitive but not arc-transitive.
The following proposition is elementary.
Proposition 5.3. For all k > 2, the only k-arc-transitive graphs of connectivity 1 are trees of constant valence.
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