K6 Minors in Large 6-Connected Graphs

Article (PDF Available) · March 2012with31 Reads
Source: arXiv
Abstract
Jorgensen conjectured that every 6-connected graph with no K_6 minor has a vertex whose deletion makes the graph planar. We prove the conjecture for all sufficiently large graphs.
K
6
MINORS IN LARGE 6-CONNECTED GRAPHS
Ken-ichi Kawarabayashi
National Institute of Informatics
2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
Serguei Norine
1
Department of Mathematics
Princeton University
Princeton, NJ 08544, USA
Robin Thomas
2
School of Mathematics
Georgia Institute of Technology
Atlanta, Georgia 30332- 0160, USA
and
Paul Wollan
Mathematisches Seminar der Universit¨at Hamburg
Bundesstrasse 55
D-20146 Hamburg, Germany
ABSTRACT
Jørgensen conjectured that every 6-connected graph G with no K
6
minor
has a vertex whose deletion makes the graph planar. We prove the conjecture
for all sufficiently large graphs.
8 April 2005, revised 22 May 2009.
1
Partially supported by NSF under Grant No. DMS-0200595.
2
Partially supported by NSF under Grants No. DMS-0200595 and. DMS-0354742.
1
1 Introduction
Graphs in this paper are allowed to have loo ps and multiple edges. A graph is a minor of
another if the first can be obtained from a subgraph of the second by contr acting edges. An
H minor is a minor isomorphic to H. A graph G is apex if it has a vertex v such that G\v
is planar. (We use \ for deletion.) Jørgensen [4] made the following beautiful conjecture.
Conjecture 1.1 Every 6-co nnected graph with no K
6
minor is apex .
This is related to Hadwiger’s conjecture [3], the following.
Conjecture 1.2 For every integer t 1, if a loopless graph has no K
t
minor, then it is
(t 1)-colorable.
Hadwiger’s conjecture is known for t 6. For t = 6 it has been proven in [12] by show-
ing that a minimal counterexample to Hadwiger’s conjecture for t = 6 is apex. The proof
uses an earlier result of Mader [6] that every minimal counterexample to Conjecture 1 .2 is
6-connected. Thus Conjecture 1.1, if true, would give more structural information. Further-
more, the structure of all graphs with no K
6
minor is not known, and appears complicated
and difficult. On the other ha nd, Conjecture 1.1 provides a nice and clean statement for
6-connected graphs. Unfortunately, it, too, appears to be a difficult problem. In this paper
we prove Conjecture 1.1 for all sufficiently large graphs, as follows.
Theorem 1.3 There exists an absolute constant N such that every 6 - connected graph on at
least N vertices w i th no K
6
minor is apex.
We use a number of results from the Graph Minor series of Robertson and Seymour,
and also three results of our own that will be proved in other papers. The first of those is
a version of Theorem 1.3 for graphs of bounded tree-width. We will not define tree-width
here, because it is sufficiently well-known, and because we do not need the concept per se,
only several theorems that use it.
Theorem 1.4 Fo r every i nteger w there exists an integer N such that every 6 -connected
graph of tree-width at most w on at least N vertices a nd with no K
6
minor is apex.
Theorem 1.4 reduces the proof of Theorem 1.3 to graphs of large tree-width. By a result of
Robertson and Seymour [8] those graphs have a larg e grid minor. However, for our purposes
it is more convenient to work with walls instead. Let h 2 be even. An elementary wall of
height h has vertex-set
{(x, y) : 0 x 2h + 1, 0 y h} {(0, 0), (2h + 1, h)}
and an edge between any vertices (x, y) and (x
, y
) if either
2
Figure 1: An elementary wall of height 4.
|x x
| = 1 and y = y
, or
x = x
, |y y
| = 1 and x and max{y, y
} have the same parity.
Figure 1 shows an element ary wall of height 4. A wall of height h is a subdivision of an
elementary wall of height h. The result of [8] (see also [2, 7, 13]) can be restated as follows.
Theorem 1.5 Fo r every eve n integer h 2 there exists a n i nteger w such that every graph
of tree-width at least w has a subgraph isomorphi c to a wall of height h.
The perimeter of a wall is t he cycle that bounds the infinite face when the wall is drawn
as in Figure 1. Now let C be the perimeter of a wall H in a graph G. The compa ss of H
in G is the restriction of G to X, where X is the union of V (C) and the vertex-set of the
unique component of G\V (C) that contains a vertex of H. Thus H is a subgraph of its
compass, and the compass is connected. A wall H with perimeter C in a graph G is planar
if its compass can be drawn in the plane with C bounding the infinite f ace. In Section 2 we
prove the following.
Theorem 1.6 Fo r every even in teger t 2 there exists an even integer h 2 such that if a
5-connected graph G with no K
6
minor has a wall of height at least h, then either it is apex,
or has a planar wall of height t.
Actually, in t he proof of Theorem 1.6 we need Lemma 2 .4 that will be proved elsewhere.
The lemma says that if a 5-connected graph with no K
6
minor has a subgraph isomorphic
to subdivision of a pinwheel with sufficiently many vanes (see Figure 2), then it is apex.
By Theorem 1.6 we may assume that our gra ph G has an arbitrarily large planar wall H.
Let C be the perimeter of H, and let K be the compass of H. Then C separates G into K
and another graph, say J, such that K J = G, V (K)V (J) = V (C) and E(K)E(J) = .
3
Next we study the graph J. Since the order of the vertices on C is import ant, we are lead
to the notion of a “society”, introduced by Robertson and Seymour in [9].
Let be a cyclic permutation of the elements of some set; we denote this set by V (Ω). A
society is a pair (G, Ω), where G is a graph, and is a cyclic permutation with V (Ω) V (G).
Now let J be as above, and let be one of the cyclic permutations of V (C) determined by
the order of vertices on C. Then (J, Ω) is a society that is of primary interest to us. We call
it the anticompass society of H in G.
We say that (G, ,
0
) is a neighborhood if G is a graph and ,
0
are cyclic permutations,
where both V (Ω) and V (Ω
0
) are subsets of V (G). Let Σ be a plane, with some orientat io n
called “ clockwise.” We say that a neighborhood (G, ,
0
) is rural if G has a drawing Γ in
Σ without crossings (so G is planar) and there are closed discs
0
Σ, such that
(i) the drawing Γ uses no point of Σ outside ∆, and none in the interior of
0
, and
(ii) for v V (G), the point of Σ representing v in the drawing Γ lies in bd(∆) (respectively,
bd(∆
0
)) if and only if v V (Ω) (respectively, v V (Ω
0
)), and the cyclic permutation of
V (Ω) (respectively, V (Ω
0
)) obtained from the clockwise orientation of bd(∆) (respectively,
bd(∆
0
)) coincides (in t he natural sense) with (respectively,
0
).
We call , Γ, ,
0
) a presentation of (G, ,
0
).
Let (G
1
, ,
0
) be a neighborhood, let (G
0
,
0
) be a society with V (G
0
)V ( G
1
) = V (Ω
0
),
and let G = G
0
G
1
. Then (G, Ω) is a so ciety, and we say that (G, Ω) is the composition
of the society (G
0
,
0
) with the neighborhood (G
1
, ,
0
). If the neighborhood (G
1
, ,
0
)
is rural, then we say that (G
0
,
0
) is a planar truncation of (G, Ω). We say that a society
(G, Ω) is k-cosmopolitan, where k 0 is an integer, if for every planar truncation (G
0
,
0
)
of (G, Ω) at least k vertices in V (Ω
0
) have at least two neighbo r s in V (G
0
). At the end of
Section 2 we deduce
Theorem 1.7 Fo r every integer k 1 there exists an even integer t 2 such that if G is
a simple graph of minimum degree at least six and H is a planar wa ll of h eight t in G, then
the an ticompass society o f H in G is k-cosmopolitan.
For a fixed presentation , Γ, ,
0
) of a neighborhood (G, ,
0
) and an integer s 0
we define an s-ne st for , Γ, ,
0
) to be a sequence (C
1
, C
2
, . . . , C
s
) of pairwise disjoint
cycles of G such that
0
1
· · ·
s
∆, where
i
denotes the closed disk in Σ
bounded by the image under Γ of C
i
. We say that a society (G, Ω) is s-nested if it is the
composition of a society (G
1
,
0
) with a rural neighborhood (G
2
, ,
0
) that has an s-nest
for some presentation o f (G
2
, ,
0
).
Let be a cyclic permutation. For x V (Ω) we denote the image of x under by
Ω(x). If X V (Ω), then we denote by |X the restriction of to X. That is, |X is the
permutation
defined by saying that V (Ω
) = X and
(x) is the first term of the sequence
Ω(x), Ω(Ω(x)), . . . which belongs to X. Let v
1
, v
2
, . . . , v
k
V (Ω) be distinct. We say that
4
(v
1
, v
2
, . . . , v
k
) is cloc kwise in (or simply clockwise when is understood from context)
if
(v
i1
) = v
i
for all i = 1, 2, . . . , k, where v
0
means v
k
and
= |{v
1
, v
2
, . . . , v
k
}. For
u, v V (Ω) we define uv as the set of all x V (Ω) such that either x = u or x = v o r
(u, x, v) is clockwise in Ω.
A separation of a graph is a pair (A, B) such that A B = V (G) and there is no edge
with one end in A B and the other end in B A. The order of (A, B) is |A B|. We say
that a society (G, Ω) is k-connected if there is no separation (A, B) of G of order at most
k 1 with V (Ω) A and B A 6= . A bump in (G, Ω) is a path in G with at least one
edge, bot h ends in V (Ω) and otherwise disjoint from V (Ω).
Let (G, Ω) be a society and let (u
1
, u
2
, v
1
, v
2
, u
3
, v
3
) be clockwise in Ω. For i = 1, 2 let P
i
be a bump in G with ends u
i
and v
i
, and let L be either a bump with ends u
3
and v
3
, or
the union of two internally disjoint bumps, one with ends u
3
and x u
3
v
3
and the other
with ends v
3
and y u
3
v
3
. In the former case let Z = , and in the latter case let Z be
the subinterval o f u
3
v
3
with ends x and y, including its ends. Assume that P
1
, P
2
, L are
pairwise disjoint. Let q
1
, q
2
V (P
1
V
2
v
3
u
3
) {u
3
, v
3
} be distinct such that neither of
the sets V (P
1
) v
3
u
1
, V (P
2
) v
2
u
3
includes both q
1
and q
2
. Let Q
1
and Q
2
be two not
necessarily disjoint paths with one end in u
3
v
3
Z {u
3
, v
3
} and the other end q
1
and q
2
,
respectively, both internally disjoint from V (P
1
P
2
L) V (Ω). In those circumstances we
say that P
1
P
2
L Q
1
Q
2
is a turtle in (G, Ω). We say that P
1
, P
2
are the legs, L is the
neck, and Q
1
Q
2
is the body o f the turtle.
Let (G, Ω) b e a society, let (u
1
, u
2
, u
3
, v
1
, v
2
, v
3
) be clockwise in Ω, and let P
1
, P
2
, P
3
be
disjoint bumps such that P
i
has ends u
i
and v
i
. In those circumstances we say that P
1
, P
2
, P
3
are three crossed paths in (G, Ω).
Let (G, Ω) be a society, and let u
1
, u
2
, u
3
, u
4
, v
1
, v
2
, v
3
, v
4
V (Ω) be such that either
(u
1
, u
2
, u
3
, v
2
, u
4
, v
1
, v
4
, v
3
) or (u
1
, u
2
, u
3
, u
4
, v
2
, v
1
, v
4
, v
3
) or (u
1
, u
2
, u
3
, v
2
= u
4
, v
1
, v
4
, v
3
) is
clockwise. For i = 1, 2, 3, 4 let P
i
be a bump with ends u
i
and v
i
such that these bumps
are pairwise disjoint, except possibly for v
2
= u
4
. In those circumstances we say that
P
1
, P
2
, P
3
, P
4
is a gridlet.
Let (G, Ω) be a society and let (u
1
, u
2
, v
1
, v
2
, u
3
, u
4
, v
3
, v
4
) be be clockwise in Ω. For
i = 1, 2, 3, 4 let P
i
be a bump with ends u
i
and v
i
such that these bumps are pairwise
disjoint, and let P
5
be a path with one end in V (P
1
) v
4
u
2
{u
2
, v
1
, v
4
}, the other end
in V (P
3
) v
2
u
4
{v
2
, v
3
, u
4
}, and otherwise disjoint from P
1
P
2
P
3
P
4
. In those
circumstances we say that P
1
, P
2
, . . . , P
5
is a separated doublec ros s .
A society (G, Ω) is rural if G can be drawn in a disk with V (Ω) drawn o n the boundary
of the disk in the order given by . A society (G, Ω) is nearly rural if there exists a vertex
v V (G) such that the society (G\v, \v) obtained from (G, Ω) by deleting v is rural.
In Sections 4–9 we prove the following. The proof strategy is explained in Section 5. It
uses a couple of theorems from [9] and Theorem 4.1 t hat we prove in Section 4.
5
Theorem 1.8 There ex i s ts an integer k 1 such that for every integer s 0 and e v ery
6-connected s-nested k-cosmopolitan society (G, Ω) either (G, ) is nearly rural, or G has a
triangle C such that (G\E(C), Ω) is rural, or (G, Ω) has an s-nested planar truncation that
has a turtle, three cross ed paths, a gridlet, or a separated doublecross.
Finally, we need to convert a turtle, three crossed paths, gridlet a nd a separated double-
cross into a K
6
minor. Let G be a 6-connected graph, let H be a sufficiently high planar wa ll
in G, and let (J, Ω) be the anticompass society of H in G. We wish to apply to Theorem 1.8
to (J, Ω). We can, in fact, assume that H is a subgraph of a larger planar wall H
that
includes s concentric cycles C
1
, C
2
, . . . , C
s
surrounding H and disjoint from H, for some
suitable integer s, and hence (J, Ω) is s-nested. Theorem 1.8 guarantees a turtle or paths
in (J, Ω) forming three crossed paths, a gridlet, or a separated double-cross, but it does not
say how the turtle or paths might intersect the cycles C
i
. In Section 10 we prove a theorem
that says that the cycles and the turtle (or paths) can be changed such that a fter po ssibly
sacrificing a lot of the cycles, the remaining cycles and the new turtle (or paths) intersect
nicely. Using t hat information it is then easy to find a K
6
minor in G. We complete the
proof of Theorem 1.3 in Section 11.
2 Finding a pl anar wall
Let a pinwheel with four vanes be the graph pictured in Figure 2. We define a pinwheel with
k vanes analogously. A graph G is internally 4-connected if it is simple, 3-connected, has at
least ve vertices, and for every separation (A, B) of G of order three, one of A, B induces
a graph with at most three edges.
Figure 2: A pinwheel with four vanes.
We assume the following terminology from [10]: distance function, perimeter, (l, m)-star
over H, external (l, m)-star over H, subwall, dividing subwall, flat subwall, society o f a wall.
6
The objective of this section is to prove the following theorem.
Theorem 2.1 Fo r every even integer t 2 there exists an even in teger h such that if H is
a wall of height at least h in an interna lly 4-connected graph G, then either
(1) G has a K
6
minor, or
(2) G has a subgraph is omorphic to a subdivision of a pinwheel with t vanes, or
(3) G has a planar wall of height t.
We begin with the following easy lemma. We leave the proof to the reader.
Lemma 2.2 For every integer t there exist integers l and m such that if a graph G has a
wall H with an external (l, m)-star, then it has a subgraph isomorphic to a pinwheel with t
vanes.
We need one more lemma, which follows immediately from [10, Theorem 8.6].
Lemma 2.3 Every flat wall in an internally 4- connected graph is planar.
Figure 3: A K
6
minor in a grid with two crosses.
Proof of Theorem 2.1. Let t 1 be given, let l, m be as in Lemma 2.2, let p = 6, and let
k, r be as in [1 0, Theorem 9.2]. If h is sufficiently large, then H has k + 1 subwalls of height
at least t, pairwise at distance at least r. If at least k of these subwalls are non- dividing,
then by [1 0, Theorem 9.2] G either has a K
6
minor, or an (l, m)-star over H, in which case
it has a subgraph isomorphic to a pinwheel with t vanes by Lemma 2.2. In either case the
theorem holds, and so we may assume that at least two of the subwalls, say H
1
and H
2
, are
dividing. We may assume that H
1
and H
2
are not planar, for otherwise the t heorem holds.
7
Let i {1, 2}. By Lemma 2.3 the wall H
i
is not flat, and hence its perimeter has a cross
P
i
Q
i
. Since the subwalls H
1
and H
2
are dividing, it follows that the paths P
1
, Q
1
, P
2
, Q
2
are pairwise disjoint. Thus G has a minor isomorphic to the graph shown in Figure 3, but
that g raph has a minor isomorphic to a minor of K
6
, as indicated by the numbers in the
figure. Thus G has a K
6
minor, and the theorem holds.
To deduce Theorem 1.6 we need the following lemma, proved in [5].
Lemma 2.4 If a 5-connected graph G with no K
6
minor has a subdivision isomorphic to a
pinwheel with 20 vanes, then G is apex .
Proof of Theorem 1.6. Let t 2 be an even integer. We may assume that t t
0
, where t
0
is as in Lemma 2.4. Let h be as in Theorem 2.1, and let G be a 5 -connected graph with no K
6
minor. From Theorem 2.1 we deduce that either G satisfies the conclusion of Theorem 1.6,
or has a subdivision isomorphic to a pinwheel with t
0
vanes. In the latter case the theorem
follows from Lemma 2.4.
We need the following theorem of DeVos and Seymour [1].
Theorem 2.5 Let (G, Ω) be a rural society such that G is a simple graph an d every ve rtex
of G not in V (Ω) has degree at least six. Then |V (G)| |V ( Ω)|
2
/12 + |V (Ω)|/2 + 1.
Proof of Theorem 1.7. Let k 1 be an integer, and let t be an even integer such that if
W is the elementar y wall of height t and |V (W )|
2
/12 + ℓ/2 + 1, then > 6k 6. Let K
be the compass of H in G, let (J, Ω) be the anticompass society of H in G, let (G
0
,
0
) be a
planar truncation of (J, Ω), and let = |V (Ω
0
)|. Thus (J, Ω) is the composition of (G
0
,
0
)
with a rural neighborhood (G
, ,
0
). Then |V (H)|
2
/12 + ℓ/2 + 1 by Theorem 2.5
applied to the society (K G
,
0
), and hence > 6k 6. Let L be the graph obtained from
K G
by a dding a new vertex v a nd joining it to every vertex of V (Ω
0
) and by adding an
edge joining every pair of nonadjacent vertices of V (Ω
0
) that are consecutive in
0
. Then L
is planar. Let s be the number of vertices of V (Ω
0
) with at least two neighbors in G
0
. Then
all but s vertices of K G
have degree in L at least six. Thus the sum of the degrees of
vertices of L is at least 6|V (K G
)| 6s + . On the other hand, the sum of the degrees is
at most 6|V (L)| 12, because L is planar, and hence s k, as desired.
3 Rural societies
If P is a path and x, y V (P ), we denote by xP y the unique subpath of P with ends x and y.
Let (G, Ω) be a society. An orderly transa ction in (G, Ω) is a sequence of k pairwise disjoint
bumps T = (P
1
, . . . , P
k
) such that P
i
has ends u
i
and v
i
and u
1
, u
2
, . . . , u
k
, v
k
, v
k1
, . . . , v
1
8
is clockwise. Let M be the graph obtained from P
1
P
2
· · · P
k
by a dding the vertices
of V (Ω) as isolated vertices. We say that M is the frame of T . We say that a path Q in G
is T -coterminal if Q has both ends in V (Ω) and is otherwise disjoint from it and for every
i = 1, 2, . . . , k the f ollowing holds: if Q intersects P
i
, then their intersection is a path whose
one end is a common end of Q and P
i
.
Let (G, Ω) be a society, and let M a nd T be as in the previous paragraph. Let i
{1, 2, . . . , k} and let Q be a T -coterminal path in G\V (P
i
) with one end in v
i
u
i
and the
other end in u
i
v
i
. In those circumstances we say that Q is a T -jump over P
i
, or simply a
T -jump.
Now let i {0, 1, . . . , k} and let Q
1
, Q
2
be two disjoint T -coterminal paths such that Q
j
has ends x
j
, y
j
and (u
i
, x
1
, x
2
, u
i+1
, v
i+1
, y
1
, y
2
, v
i
) is clockwise in Ω, where possibly u
i
= x
1
,
x
2
= u
i+1
, v
i+1
= y
1
, or y
2
= v
i
, and u
0
means x
1
, u
k+1
means x
2
, v
k+1
means y
1
, and v
0
means y
2
. In those circumstances we say that (Q
1
, Q
2
) is a T -cross in region i, or simply a
T -cross.
Finally, let i {1, 2, . . . , k} and let Q
0
, Q
1
, Q
2
be three paths such that Q
j
has ends
x
j
, y
j
and is otherwise disjoint from all members of T , x
0
, y
0
V (P
i
), the vertices x
1
, x
2
are
internal vertices of x
0
P
i
y
0
, y
1
, y
2
6∈ V (P
i
), y
1
u
i1
u
i
v
i
v
i1
, y
2
u
i
u
i+1
v
i+1
v
i
, and
the paths Q
0
, Q
1
, Q
2
are pa irwise disjoint, except possibly x
1
= x
2
. In those circumstances
we say that (Q
0
, Q
1
, Q
2
) is a T -tunnel under P
i
, or simply a T -tunnel.
Intuitively, if we think of the paths in T as dividing the society into “regions”, then
a T -jump arises from a T -path whose ends do not belong to the same region. A T -cross
arises from two T -paths with ends in the same region that cross inside that region, and
furthermore, each path in T includes at most two ends of those crossing paths. Finally,
a T -tunnel can be converted into a T -jump by rerouting P
i
along Q
0
. However, in some
applications such rerouting will be undesirable, and therefore we need to list T -tunnels as
outcomes.
Let M be a subgraph of a graph G. An M-bridg e in G is a connected subgraph B of G
such that E(B) E(M) = and either E(B) consists of a unique edge with both ends in M,
or for some component C of G\V (M) the set E(B) consists of all edges of G with at least
one end in V (C). The vertices in V (B) V (M) are called the attachments of B. Now let M
be such that no block of M is a cycle. By a s egment of M we mean a maximal subpath P of
M such that every internal vertex of P has degree two in M. It follows that the segments of
M are uniquely determined. Now if B is an M-bridge of G, then we say that B is unstable if
some segment of M includes all the attachments of B, and otherwise we say that B is stable.
A so ciety (G, Ω) is rurally 4-connected if for every separation (A, B) of o r der at most
three with V (Ω) A the graph G[B] can be drawn in a disk with the vertices of A B
drawn on the boundary of the disk. A society is cross-free if it has no cross. The following,
a close relative of Lemma 2.3, follows from [9, Theorem 2.4].
9
Theorem 3.1 Every cross-free rurally 4-co nnected society is rural.
Lemma 3.2 Let (G, Ω) be a rurally 4-connected society, let T = (P
1
, . . . , P
k
) be an o rde rly
transaction in (G, Ω), and let M be the f rame of T . If every M-bridge of G is stable and
(G, Ω) is not rural, then (G, Ω) has a T -jump , a T -cross, or a T -tunnel.
Proof. For i = 1, 2, . . . , k let u
i
and v
i
be the ends of P
i
numbered as in t he defintion
of orderly transaction, and for convenience let P
0
and P
k+1
be null graphs. We define
k + 1 cyclic permutations
0
,
1
, . . . ,
k
as follows. For i = 1, 2, . . . , k 1 let V (Ω
i
) :=
V (P
i
) V (P
i+1
) u
i
u
i+1
v
i+1
v
i
with t he cyclic o r der defined by saying that u
i
u
i+1
is followed by V (P
i+1
) in order fr om u
i+1
to v
i+1
, followed by v
i+1
v
i
followed by V (P
i
) in
order from v
i
to u
i
. The cyclic permutation
0
is defined by letting v
1
u
1
be followed by
V (P
1
) in order from u
1
to v
1
, and
k
is defined by letting u
k
v
k
be followed by V (P
k
) in
order from v
k
to u
k
.
Now if for some M-bridge B of G there is no index i {0, 1, . . . , k} such that all
attachments of B belong to V (Ω
i
), then (G, Ω) has a T -jump. Thus we may assume that
such index exists for every M-bridge B, and since B is stable that index is unique. Let us
denote it by i(B). For i = 0, 1, . . . , k let G
i
be the subgraph of G consisting of P
i
P
i+1
,
the vertex-set V (Ω
i
) and all M-bridges B of G with i(B) = i. The society (G
i
,
i
) is rurally
4-connected. If each (G
i
,
i
) is cross-free, then each of t hem is rural by Theorem 3.1 and it
follows that (G, Ω) is rural. Thus we may assume that for some i = 0, 1, . . . , k the society
(G
i
,
i
) has a cross (Q
1
, Q
2
). If neither P
i
nor P
i+1
includes three or four ends of the paths
Q
1
and Q
2
, then (G, Ω) has a T -cross. Thus we may assume that P
i
includes both ends
of Q
1
and at least one end of Q
2
. Let x
j
, y
j
be the ends of Q
j
. Since the M-bridge of G
containing Q
2
is stable, it has an attachment outside P
i
, and so if needed, we may replace
Q
2
by a path with an end outside P
i
(or conclude that (G, Ω) has a T -jump). Thus we may
assume that u
i
, x
1
, x
2
, y
1
, v
i
occur on P
i
in the order listed, and y
2
6∈ V (P
i
).
The M-bridge of G containing Q
1
has an attachment outside P
i
. If it does not include
Q
2
and has an attachment outside V (P
i
) {y
2
}, then (G, Ω) has a T -j ump or T -cross, and
so we may assume not. Thus there exists a path Q
3
with one end x
3
in the interior of Q
1
and the o t her end y
3
V (Q
2
) {x
2
} with no internal vertex in M Q
1
Q
2
. We call the
triple (Q
1
, Q
2
, Q
3
) a tripod, and the path y
3
Q
2
y
2
the leg of the tripod. If v is an internal
vertex of x
1
P
i
y
1
, then we say that v is sheltered by the tripod (Q
1
, Q
2
, Q
3
). Let L be a path
that is the leg of some tripod, and subject to that L is minimal. From now on we fix L and
will consider different tripods with leg L; t hus the vertices x
1
, y
1
, x
2
, x
3
may change, but y
2
and y
3
will remain fixed as the ends of L.
Let x
1
, y
1
V (P
i
) be such that they are sheltered by no tripod with leg L, but every
internal vertex of x
1
P
i
y
1
is sheltered by some tripod with leg L. Let X
be the union
of x
1
P
i
y
1
and all tripods with leg L that shelter some internal vertex of x
1
P
i
y
1
, let X :=
10
X
\V (L)\{x
1
, y
1
} and let Y := V (M L)x
1
P
i
y
1
{y
3
}. Since (G, Ω) is rurally 4-connected
we deduce that the set {x
1
, y
1
, y
3
} does not separate X from Y in G. It follows that there
exists a path P in G\{x
1
, y
1
, y
3
} with ends x X and y Y . We may assume that P has
no internal vertex in X Y . Let (Q
1
, Q
2
, Q
3
) be a tripod with leg L such that either x is
sheltered by it, or x V (Q
1
Q
2
Q
3
). If y 6∈ V (L P
i
), then by considering the paths
P, Q
1
, Q
2
, Q
3
it follows that either (G, Ω) has a T - jump or T - t unnel. If y V (L), t hen there
is a tripod whose leg is a proper subpath of L, contrary to the choice of L. Thus we may
assume t hat y V (P
i
), and that y V (P
i
) for every choice of the path P as above. If
x V (Q
1
Q
2
Q
3
) then there is a tripod with leg L that shelters x
1
or y
1
, a contradiction.
Thus x V (P
i
). Let B be the M-bridge containing P . Since y V (P
i
) for all choices of
P it follows that the attachments of B are a subset of V (P
i
) {y
2
}. But B is stable, and
hence y
2
is an attachment of B. The minimality of L implies that B includes a path from y
to y
3
, internally disjoint from L. Using that path and the paths P, Q
1
, Q
2
, Q
3
it is now easy
to construct a tripod that shelters either x
1
or y
1
, a contradiction.
4 Leap of length fi ve
A leap of l ength k in a society (G, Ω) is a sequence of k + 1 pairwise disjoint bumps
P
0
, P
1
, . . . , P
k
such that P
i
has ends u
i
and v
i
and u
0
, u
1
, u
2
, . . . , u
k
, v
0
, v
k
, v
k1
, . . . , v
1
, is
clockwise. In this section we prove the following.
Theorem 4.1 Let (G, Ω) be a 6 - connected society with a lea p of length five. Then (G, )
is nearly rural, or G has a triangl e C such that (G\E(C), Ω) is rural, o r (G, Ω) h as three
crossed paths, a gridlet, a separated doublecross, or a turtle.
The following is a hypothesis that will be common to several lemmas of this section, and
so we state it separately to avoid repetition.
Hypothesis 4.2 Let (G, Ω) be a society with no three crossed paths, a gridlet, a separated
doublecross, or a turtle, let k 1 be an integer, let
(u
0
, u
1
, u
2
, . . . , u
k
, v
0
, v
k
, v
k1
, . . . , v
1
)
be clockwise, and let P
0
, P
1
, . . . , P
k
be pairwise disjoint bumps such that P
i
has ends u
i
and
v
i
. Let T be the orderly transaction (P
1
, P
2
, . . . , P
k
), let M be the frame of T and let
Z = u
1
u
k
v
k
v
1
V (P
2
) V (P
3
) · · · V (P
k1
) {u
1
, u
k
, v
1
, v
k
}.
Let Z
1
= v
1
u
1
{u
0
, u
1
, v
1
} and Z
2
= u
k
v
k
{v
0
, u
k
, v
k
}.
11
If H is a subgraph of G, then an H-path is a path of length at least one with both ends
in V (H) and otherwise disjoint from H. We say that a vertex v of P
0
is exposed if there
exists an (M P
0
)-path P with one end v and the other in Z.
Lemma 4.3 Assume Hypothesis 4.2 and let k 3. Let R
1
, R
2
be two disjoint (M P
0
)-
paths in G such that R
i
has ends x
i
V (P
0
) and y
i
V (M) {u
0
, v
0
}, and a s sume that
u
0
, x
1
, x
2
, v
0
occur on P
0
in the order listed, where possibly u
0
= x
1
, or v
0
= x
2
, or both.
Then e i ther y
1
V (P
1
) v
1
u
1
, or y
2
V (P
k
) u
k
v
k
, or both. In particular, there do not
exist two disjoint (M P
0
)-paths from V (P
0
) to Z.
Proof. The second statement follows immediately from the first, and so it suffices to prove
the first statement. Suppose for a contradiction that there exist pa ths R
1
, R
2
satisfying the
hypotheses but not the conclusion of the lemma. By using the paths P
2
, P
3
, . . . , P
k1
we
conclude that there exist two disjoint paths Q
1
, Q
2
in G such that Q
i
has ends x
i
V (P
0
)
and z
i
V (Ω), and is otherwise disjoint from V (P
0
) V (Ω), and if Q
i
intersects some P
j
for
j {1 , 2, . . . , k}, then j {2, . . . , k 1} and Q
i
P
j
is a path one of whose ends is a common
end of Q
i
and P
j
. Furthermore, z
1
u
1
v
1
{u
1
, v
1
} and z
2
v
k
u
k
{u
k
, v
k
}. From
the symmetry we may assume that either (u
0
, v
0
, z
2
, z
1
), or (u
0
, z
1
, v
0
, z
2
) or (u
0
, v
0
, z
1
, z
2
) is
clockwise. In the first two cases (G, Ω) has a separated doublecross (the two pairs of crossing
bumps are P
1
and Q
1
u
0
P
0
x
1
, and P
k
and Q
2
v
0
P
0
x
2
, and the fifth path is a subpath
of P
2
), unless the second case holds and z
1
u
k
v
0
or z
2
v
1
u
0
, or both. By symmetry
we may assume that z
1
u
k
v
0
. Then, if z
2
v
k2
u
0
, (G, Ω) has a gridlet formed by
the paths P
k
, P
k1
, u
0
P
0
x
1
Q
1
and v
0
P
0
x
2
Q
2
. Otherwise, z
2
v
k
v
k2
{v
k
, v
k2
} and
(G, Ω) has a turtle with legs P
k
and v
0
P
0
x
2
Q
2
, neck P
1
and body u
0
P
0
x
2
Q
1
.
Finally, in the third case (G, Ω) has a turtle or three crossed paths. More precisely, if
z
2
v
0
v
1
{v
1
}, then (G, Ω) has a turtle described in the paragraph above. Otherwise,
by symmetry, we may assume that z
2
v
1
u
0
and z
1
v
0
v
k
, in which case v
0
P
0
x
2
Q
2
,
u
0
P
0
x
1
Q
1
and P
2
are t he three crossed paths.
Lemma 4.4 Assume Hypothesis 4.2 and l et k 2. Then (G\V (P
0
), \V (P
0
)) has no T -
jump.
Proof. Suppose fo r a contradiction that (G\V (P
0
), \V (P
0
)) has a T - jump. Thus there is
an index i {1, 2, . . . , k} and a T -coterminal path P in G\V (P
0
P
i
) with ends x v
i
u
i
and y u
i
v
i
. Let j {1, 2, . . . , k} {i}. Then using the paths P
0
, P
i
, P
j
and P we deduce
that (G, Ω) has either three crossed pat hs or a gridlet, in either case a contradiction.
Lemma 4.5 Assume Hypothesis 4 . 2 and let k 2. Let v V (P
0
) be such that there is no
(M P
0
)-path in G\v from vP
0
v
0
to vP
0
u
0
V (P
1
P
2
· · · P
k1
) v
k
u
k
{v
k
, u
k
}
12
and none from vP
0
u
0
to V (P
2
P
3
· · · P
k
) u
1
v
1
{u
1
, v
1
}. Then (G \v, \v) has no
T -jump.
Proof. The hypotheses of the lemma imply that every T -jump in (G\v, \v) is disjoint
from P
0
. Thus the lemma follows from Lemma 4.4.
Lemma 4.6 Assume Hypothesis 4.2, let k 3, and let v V (P
0
) be such that no vertex in
V (P
0
) {v} is expos ed. Let i {0, 1, . . . , k} be such that (G\v, \v) has a T -cross (Q
1
, Q
2
)
in region i. Then i {0, k} and v is not exposed. Furthermore, ass ume that i = 0 , and that
there exists an (M P
0
)-path Q with one end v and the other end in P
1
v
1
u
1
{u
0
}, and
that v
0
P
0
v is disjoint from Q
1
Q
2
. Then for some j {1, 2} there exi s t p V (Q
j
u
0
P
0
v)
and q V (Q
j
Q) such that pP
0
v and qQv are internally disjoint from Q
1
Q
2
.
Proof. If i 6∈ {0, k}, then the T -cross is disjoint from P
0
by the choice of v, and hence the
T -cross and P
0
give rise to three crossed paths. To complete the proof of the first assertion
we may assume that i = 0 and that v is exposed. Thus there exists a T -coterminal path
Q
from v to Z V (Ω) disjoint from P
0
P
1
P
k
\v. If (Q
vP
0
v
0
) (Q
1
Q
2
) = then
(G, Ω) has a separated doublecross, where one pair of crossed paths is obtained f r om the
T -cross, the other pair is P
k
and Q
vP
0
v
0
, and the fifth path is a subpath of P
2
. Thus
we may assume that there exists x (V (Q
′′
)) V (Q
1
) and that x is chosen so t hat xQ
′′
y is
internally disjoint from Q
1
Q
2
, where Q
′′
= Q
vP
0
v
0
and y is t he end of Q
in Z V (Ω).
Let x
(V (P
0
) (V (Q
1
) V (Q
2
)) {u
0
} be chosen so that x
P
0
v
0
is internally disjoint
from Q
1
Q
2
. Let z
1
v
1
u
1
{v
1
, u
1
} be an end of Q
1
. If x V (Q
), then Q
1
is disjoint
from P
0
, because v is the only exposed vertex. Thus z
1
Q
1
x xQ
y is a T -jump disjoint
from P
0
, contrary to Lemma 4.4. It follows that x V (v
0
P
0
v), and Q
is disjoint from
Q
1
Q
2
. Let j { 1, 2} be such that x
V (Q
j
), let z
j
v
1
u
1
{v
1
, u
1
} be an end of Q
j
and let P
0
:= v
0
P
0
x
x
Q
j
z
j
. If x
Q
j
z
j
does not intersect u
0
P
0
v, then u
0
P
0
v Q
is a T -
jump, disjoint from P
0
, contrary to Lemma 4.4; otherwise there exist two pa ths contradicting
Lemma 4.3 applied to T a nd the path P
0
: one is a subpath of Q
j
and the other is a subpath
of u
0
P
0
v Q
. This proves the first assertion of the lemma.
To prove the second statement of the lemma we assume that i = 0 and that Q is a path
from v to v
v
1
u
1
{u
0
}, disjoint from M P
0
\v, except that P
1
Q may be a path
with one end v
. Let the ends of Q
1
, Q
2
be labeled as in the definition of T -cross. If P
0
is disjoint from Q
1
Q
2
, then (G, Ω) has three crossed paths (if (y
2
, u
0
, x
1
) is clockwise) or
a gridlet with paths Q
1
, Q
2
, P
0
, P
2
(if (x
1
, u
0
, x
2
) or (y
1
, u
0
, y
2
) is clockwise), or a separated
doublecross with paths Q
1
, Q
2
, P
0
, P
2
, P
k
(if (v
1
, u
0
, y
1
) or (x
2
, u
0
, u
1
) is clockwise). Thus we
may assume t hat P
0
intersects Q
1
Q
2
. (Please note t hat v
0
P
0
v is disjoint from Q
1
Q
2
by hypothesis.) Similarly we may assume that Q intersects Q
1
Q
2
, for otherwise we apply
the previous argument with P
0
replaced by Q vP
0
v
0
. Let p V (Q
1
Q
2
) u
0
P
0
v and
13
q V (Q
1
Q
2
) V (Q) be chosen to minimize pP
0
v and qQv. If p and q belong to different
paths Q
1
, Q
2
, t hen (G, Ω) has a turtle with legs Q
1
, Q
2
, neck P
k
and body pP
0
v
0
qQv. Thus
p and q belong to the same Q
j
and the lemma holds.
In the proof o f the following lemma we will be applying Lemma 3.2. To guarantee t hat
the conditions of Lemma 3.2 are satisfied, we will need a result from [5]. We need to precede
the statement of this result by a few definitions.
Let M be a subgraph of a graph G, such that no block of M is a cycle. Let P be a
segment of M of length at least two, and let Q be a path in G with ends x, y V (P ) and
otherwise disjoint from M. Let M
be obtained from M by replacing the path xP y by Q;
then we say that M
was obtained from M by rerouting P along Q, or simply that M
was
obtained from M by rerouting. Please note that P is required to have length at least two,
and hence this relation is not symmetric. We say that the rerouting is p roper if all the
attachments of the M-bridge that contains Q belong to P . The following is proved in [5 ,
Lemma 2.1].
Lemma 4.7 Let G be a graph, and let M be a subgraph of G such that no block of M is
a cycle. Then there exists a subgraph M
of G obtained from M by a sequence of proper
reroutings such that if an M
-bridge B of G is unstable, say all its attachments belong to
a segment P of M
, then there exist vertices x, y V (P ) such that some component of
G\{x, y} i ncludes a vertex of B and is disjoint from M\V (P ).
Lemma 4.8 Assume Hypothesis 4.2, and let k 4. If every leap of length k 1 has at most
one exposed vertex, (G, Ω) is 4 - connected and (G\v, \v) is rurally 4-connected for every
v V (P
0
), then (G, Ω) is nearly rural.
Proof. Since (G, Ω) has no separated doublecross it follows that it does not have a T -cross
both in region 0 and region k. Thus we may assume that it has no T -cross in region k.
Similarly, it f ollows that it does not have a T -tunnel under both P
1
and P
k
, or a T -cross in
region 0 and a T -tunnel under P
k
. Thus we may also assume that (G, Ω) has no T -tunnel
under P
k
. If some leap of length k in (G, Ω) has an exposed vertex, then we may assume
that v is an exposed vertex. Otherwise, let the leap (P
0
, P
1
, . . . , P
k
) and v V (P
0
) be chosen
such that either v = u
0
or there exists an (M P
0
)-path with one end v and the other end
in P
1
v
1
u
1
{u
0
}, and, subject to that, vP
0
v
0
is as short as possible.
By Lemma 4.7 we may assume, by properly rerouting M if necessary, that every M-bridge
of G\v is stable. Since the reroutings are proper the new paths P
i
will still be disjoint from
P
0
, and the property that defines v will continue to hold. Similarly, the facts that there is no
T -cross in region k and no T -tunnel under P
k
remain unaffected. We claim that v satisfies
the lemma.
14
We apply Lemma 3.2 to the society (G\v, \v) and orderly tr ansaction T . We may
assume that (G\v, \v) is not rural, a nd hence by Lemma 3.2 the society (G\v, \v) has
a T - jump, a T -cross or a T -tunnel. By the choice of v there exists a path Q from v to
v
v
k
u
k
{ v
k
, u
k
} such that Q does not intersect P
k
P
0
\v and intersects at most one of
P
1
, P
2
, . . . , P
k1
. Furthermore, if it intersects P
i
for some i {1, 2, . . . , k 1} then P
i
Q is
a path with one end a common end of both.
We claim that v satisfies the hypotheses of Lemma 4.5. To prove this claim suppose for
a contradiction tha t P is an (M P
0
)-path violating that hypothesis. Suppose first that P
and Q ar e disjoint. Then P joins different components of P
0
\v by Lemma 4.3. But then
changing P
0
to the unique path in P
0
P that does not use v either produces a leap with at
least two exposed vertices, or contradicts the minimality of vP
0
v
0
. Thus P and Q intersect.
Since no leap of length k has two or more exposed vertices, it follows that v is not exposed.
Thus P has one end in u
0
P
0
v by the minimality of vP
0
v
0
, and the other end in P
k
u
k
v
k
,
because v is not exp osed. But t hen P Q includes a T -jump disjoint from P
0
, contrary
to Lemma 4.4. This proves our claim that v satisfies the hypotheses of Lemma 4.5. We
conclude that (G\v, \v) has no T -jump.
Assume now that (G\v, \v) has a T -cross (Q
1
, Q
2
) in region i. Then by the first part o f
Lemma 4.6 and the assumption made earlier it follows that i = 0 and v is not exposed. But
the existence of Q and the second statement of Lemma 4.6 imply that some leap of length k
has at least two exposed vertices, a contradiction. (To see that let j, p, q be as in Lemma 4.6.
Replace P
1
by Q
3j
and replace P
0
by a suitable subpath of Q
j
pP
0
v
0
qQv.)
We may therefore assume that (G\v, \v) has a T -tunnel (Q
0
, Q
1
, Q
2
) under P
i
for some
i {1, 2, . . . , k}. Then the leap L
= (P
0
, P
1
, . . . , P
i1
, P
i+1
, . . . , P
k
) of length k 1 3 has
a T
-cross, where T
is the corresponding orderly society, and the result follows in the same
way as above.
Lemma 4.9 Assume Hypothesis 4.2 and let k 3. If there e xist at least two exposed
vertices, then there exists a cycle C and three disjoint (M C)-paths R
1
, R
2
, R
3
such that
R
i
has ends x
i
V (C) and y
i
V (M), C\{x
1
, x
2
, x
3
} is disjoint from M, y
1
= u
0
, y
2
= v
0
and y
3
Z.
Proof. Let x
1
be the closest exposed vertex to u
0
on P
0
, and let x
2
be the closest exposed
vertex to v
0
. Let R
1
= P
0
[x
1
, u
0
] and let R
2
= P
0
[x
2
, v
0
]. For i = 1, 2 let S
i
be an (M P
0
)-
path with one end x
i
and the other end in Z. By Lemma 4.3 S
1
and S
2
intersect, and so
we may assume that S
1
S
2
is a path R
3
containing an end of both S
1
and S
2
, say y
3
. Let
x
3
be the other end of R
3
. Then P
0
S
1
S
2
includes a unique cycle C. The cycle C and
paths R
1
, R
2
, R
3
are as desired for the lemma.
15
If the cycle C in Lemma 4.9 can be chosen to have at least four vertices, then we say
that the leap (P
0
, P
1
, . . . , P
k
) is diverse.
Lemma 4.10 Assume Hypothesis 4.2, l et k 4, and let there be no diverse leap o f le ngth
k. If C is as in Lemma 4.9 and (G\E(C), Ω) is rurally 4-connected, then (G\E(C), ) is
rural.
Proof. Since the leap (P
0
, P
1
, . . . , P
k
) is not diverse, it f ollows that C is a triangle. Let
R
1
, R
2
, R
3
and their ends be numbered as in Lemma 4.9. We may assume that P
0
= R
1
R
2
+ x
1
x
2
. Since t here is no diverse leap, Lemma 4.3 implies that there is no path in
G\E(C)\V (P
k
) from x
2
to v
k
u
k
, and no ne in G\E(C)\V (P
1
) from x
1
to u
1
v
1
. It also
implies that no vertex on P
0
is exp osed in G\x
1
x
3
\x
2
x
3
.
As in Lemma 4.8, we can apply L emma 4.7 and assume, by properly rerouting M if neces-
sary, that the conditions of Lemma 3.2 are satisfied. We assume that the society (G\E(C), Ω)
has a T -jump, a T -cross, or a T -tunnel, as otherwise by Lemma 3.2 (G\E(C), Ω) is rural.
By the observation at the end of the previous paragraph this T -jump, T -cross, or T -tunnel
cannot use both x
1
and x
2
; say it does not use x
2
. But that contradicts Lemma 4.5 or the
first part of Lemma 4.6, applied to v = x
2
and the graph G\x
1
x
3
, in case of a T -jump or a
T -cross.
Thus we may assume that (G\E(C)\x
2
, \x
2
) has a T -tunnel (Q
0
, Q
1
, Q
2
) under P
i
for
some i {1, 2, . . . , k}. But then the leap L
= (P
0
, P
1
, . . . , P
i1
, P
i+1
, . . . , P
k
) of length
k 1 3 has a T
-cross (Q
1
, Q
2
), where T
is the corresponding orderly transaction, Q
1
is
obtained from P
i
by rerouting along Q
0
and Q
2
is the union of Q
1
Q
2
with the subpath
of P
i
joining the ends of Q
1
and Q
2
. By the first half of Lemma 4.6 applied to the graph
G\x
1
x
3
, the leap L
, v := x
2
and the T
-cross (Q
1
, Q
2
) we may assume that i = 1 and
that y
3
v
2
u
2
{u
0
}. By the second half of Lemma 4.6 a pplied to the same entities and
Q := R
3
+ x
3
x
2
there exist j {1 , 2}, p V (Q
j
R
1
) and q V (Q
j
Q) such that pP
0
x
2
and qQx
2
are internally disjoint from Q
1
Q
2
. If j = 1, then p, q belong to the interior of Q
0
,
and the leap (P
0
, P
1
, . . . , P
k
) is diverse, as a subpath of Q
0
joins a vertex of R
1
to a vertex
of Q in G\x
1
x
3
. If j = 2 then we obtain a diverse leap from (P
0
, P
1
, . . . , P
k
) by replacing P
1
by Q
1
and replacing P
0
by a suitable subpath of Q v
0
P
0
p Q
2
.
Lemma 4.11 Assume Hypothesis 4.2, let k 3, let (G, Ω) be 4-conn ected, l et C, R
1
, R
2
, R
3
be as in Lemma 4.9, and assume that C is not a triangle. Then there exist four disjoint
(M C)-paths, each with one end in V (C) and the other end respectively in the sets { u
0
},
{v
0
}, Z and V (P
1
P
k
).
Proof. By an application of the proof of the max-flow min-cut theorem there exist four
disjoint (M C)-paths, each with one end in V (C) and t he o t her end respectively in t he sets
16
{u
0
}, {v
0
}, Z and V (M). By Lemma 4.3 the fourth path does not end in V (M) V (P
1
)
V (P
k
). The result follows.
Lemma 4.12 Assume Hypothesis 4.2, le t k 3, let C, R
1
, R
2
, R
3
be as i n Lemma 4.9, let
D := M C R
1
R
2
R
3
, and let R
4
be a D-path with ends x
4
V (C) {x
1
, x
2
, x
3
}
and y
4
V (P
1
). Then x
1
, x
2
, x
3
, x
4
occur on C in the order listed. Furthermore, if R is a
D-path with ends x V (C) {x
1
, x
2
, x
3
} and y V (M), then x
1
, x
2
, x
3
, x occur o n C in
the order listed and y V (P
1
).
Proof. The vertices x
1
, x
2
, x
3
, x
4
occur on C in the order listed by Lemma 4.3. Now let R
be as stated. By Lemma 4.3 we have y V (P
1
P
k
), and so by the first par t of the lemma
we may assume that y V (P
k
). By the symmetric statement to the first half of the lemma
it fo llows that x
1
, x
2
, x, x
3
occur on C in the order listed. We may assume that P
0
is the
unique path f r om u
0
to v
0
in R
1
R
2
C\x
3
. Then R
4
R C\V (P
0
) includes a T -jump
disjoint from P
0
, contrary to L emma 4.4.
We need to further upgra de the assumptions of Hypothesis 4.2, as follows.
Hypothesis 4.13 Assume Hypothesis 4.2. Let C be a cycle with distinct vertices x
1
, x
2
, x
3
such that C\{x
1
, x
2
, x
3
} is disjoint from M. Let R
1
, R
2
, R
3
be pairwise disjoint (M C)-
paths such that R
i
has ends x
i
and y
i
, where y
1
= u
0
, y
2
= v
0
, and y
3
Z. By a ray we mean
an (M C)-path from C to M, disjoint from R
1
R
2
R
3
. We say that a vertex v V (P
1
)
is ill umi nated if there is a ray with end v. Let x
4
, x
5
V (P
1
) be illuminated vertices such
that either x
4
= x
5
, or u
1
, x
4
, x
5
, v
1
occur on P
1
in the order listed, and x
4
P
1
x
5
includes all
illuminated vertices. Let R
4
:= u
1
P
1
x
4
and R
5
:= v
1
P
1
x
5
, and let y
4
:= u
1
and y
5
:= v
1
.
Let S
4
and S
5
be rays with ends x
4
and x
5
, respectively, and let A
0
:= V (M) V (P
1
) and
B
0
:= V (C S
4
S
5
x
4
P
1
x
5
).
Lemma 4.14 Assume Hypothesis 4.13 , let k 3, and let (G, Ω) be 6-connec ted. Then
x
4
6= x
5
, and the path x
4
P
1
x
5
has at least one internal vertex.
Proof. If x
4
= x
5
or x
4
P
1
x
5
has no internal vertex, then by Lemma 4.12 the set {x
1
, x
2
, . . . , x
5
}
is a cutset separating C from M\V (P
1
), contrary to the 6-connectivity of (G, Ω). Note that
V (C) {x
1
, x
2
, . . . , x
5
} is non-empty as it includes an end of a ray.
Assume Hypothesis 4 .1 3. By Lemma 4.14 the paths R
1
, R
2
, . . . , R
5
are disjoint paths
from A
0
to B
0
. The following lemma follows by a standard “augmenting path” arg ument.
Lemma 4.15 Assume Hypothesis 4.13, and let k 2. I f there is no separation (A, B) of
order at most five with A
0
A and B
0
B, then there e x ist an integer n and internally
disjoint paths Q
1
, Q
2
, . . . , Q
n
in G, where Q
i
has distinct ends a
i
and b
i
such that
17
(i) a
1
A
0
{y
1
, y
2
, . . . , y
5
} a nd b
n
B
0
{x
1
, x
2
, . . . , x
5
},
(ii) f o r all i = 1, 2, . . . , n 1, a
i+1
, b
i
V (R
t
) for some t {1, 2, . . . , 5}, and y
t
, a
i+1
, b
i
, x
t
are pairwise distinct and occur on R
t
in the order listed,
(iii) if a
i
, b
j
V (R
t
) for some t {1, 2, . . . , 5} and i, j {1, 2, . . . , 5} with i > j + 1, then
either a
i
= b
j
, or y
t
, b
j
, a
i
, x
t
occur on R
t
in the order listed, and
(iv) for i = 1, 2, . . . , n, if a v ertex of Q
i
belongs to A
0
B
0
V (R
1
R
2
· · · R
5
), then
it is an end o f Q
i
.
The sequence of paths (Q
1
, Q
2
, . . . , Q
n
) as in Lemma 4.15 will b e called an augmenting
sequence.
Lemma 4.16 Assume Hypothesis 4.13, and let k 3. Then there is no augmenting sequence
(Q
1
, Q
2
, . . . , Q
n
), where Q
1
is disjoint from P
2
.
Proof. Suppose for a contradiction that there is an augmenting sequence (Q
1
, Q
2
, . . . , Q
n
),
where Q
1
is disjoint from P
2
, and let us assume t hat the leap (P
0
, P
1
, . . . , P
k
), cycle C, paths
R
1
, R
2
, R
3
, S
4
, S
5
and augmenting sequence (Q
1
, Q
2
, . . . , Q
n
) are chosen with n minimum.
Let the ends of the paths Q
i
be labeled as in Lemma 4.15. We may assume that P
0
is the
unique path from u
0
to v
0
in R
1
R
2
C\x
3
. We proceed in a series of claims.
(1) The vertex b
n
belongs to the interior of x
4
P
1
x
5
.
To prove (1) suppose for a contradiction that b
n
V (C S
4
S
5
). By L emma 4 .1 2, the
choice of x
4
, x
5
and the fact that a
n
6= x
4
, x
5
by Lemma 4.15(ii) we deduce that a
n
V (R
i
)
for some i {1, 2, 3}. Then we can use Q
n
to modify C to include a
n
R
i
x
i
(and modify
R
1
, R
2
, R
3
accordingly), in which case (Q
1
, Q
2
, . . . , Q
n1
) is an augmentation contradicting
the choice of n. This proves (1).
(2) a
i
, b
i
V (R
j
) for no i {1, 2, . . . , n} and no j {1, 2, . . . , 5}.
To prove (2) suppose to the contrary that a
i
, b
i
V (R
j
). Then 1 < i < n and by
rerouting R
j
along Q
i
we obtain an augmentatio n (Q
1
, Q
2
, . . . , Q
i2
, Q
i1
b
i1
R
j
a
i+1
Q
i+1
, Q
i+2
, . . . , Q
n
), contrary to the minimality of n. This proves (2).
(3) a
i
, b
i
V (R
1
R
2
R
3
) for no i {1, 2, . . . , n}.
Using (2) the proof of (3) is analogous to the argument at the end of the proof o f Claim ( 1).
(4) a
i
, b
i
V (R
4
R
5
) for no i {1, 2, . . . , n}.
By (2) one of a
i
, b
i
belongs to R
4
and the o t her to R
5
. We can reroute P
1
along Q
i
, and then
(Q
1
, Q
2
, . . . , Q
i1
) becomes an augmentation, contrary to the minimality of n.
18
(5) For i = 1, 2, . . . , n 1, the graph Q
i
R
1
R
2
R
3
includes no T -jump.
This claim follows from (3), Lemma 4.3 and Lemma 4.4 a pplied to P
0
.
(6) a
1
6∈ v
1
u
1
.
To prove (6) suppose for a contr adiction that a
1
v
1
u
1
. Since a
1
6= y
1
, we may assume
from the symmetry t hat a
1
v
1
y
1
{ y
1
}. Then b
1
V (P
1
R
1
) by (5). But if b
1
V (R
i
),
where i = 1 or i = 5, then by rerouting R
i
along Q
1
we obtain an augmenting sequence
(Q
2
x
1
R
i
a
2
, Q
3
, Q
4
, . . . , Q
n
), contrary to the choice of n. Thus b
1
u
1
P
1
x
5
. By replacing
P
1
by the path Q
1
u
1
P
1
b
1
and considering the paths R
3
and S
5
R
5
we obtain contradiction
to Lemma 4.3. This proves (6).
(7) a
1
6∈ u
k
v
k
.
Similarly as in the proof of (6), if a
1
u
k
v
k
, then b
1
V (R
2
) by (5), and we reroute
R
2
along Q
1
to obtain a contra diction to the minimality of n. This proves (7).
(8) a
1
V (P
k
).
To prove (8) we may assume by ( 6) and (7) that a
1
Z. Then b
1
V (R
3
P
1
) by (5).
If b
1
V (R
3
), then we reroute R
3
along Q
1
as before. Thus b
1
V (P
1
). It follows from ( 5)
and the hypo t hesis V (P
2
) V (Q
1
) = t hat a
1
u
1
u
2
{u
1
, u
2
} or a
1
v
2
v
1
{v
1
, v
2
},
and so from the symmetry we may assume the latter.
Let us assume for a moment that y
3
a
1
v
1
. We reroute P
1
along Q
1
b
1
P
1
v
1
. The union
of R
3
, R
2
and a path in C between x
2
and x
3
, avoiding x
1
, x
4
, x
5
, will play the role of P
0
after rerouting. If b
1
x
4
P
1
v
1
{x
4
}, then R
1
C S
4
R
4
includes two disjoint paths that
contradict Lemma 4.3 applied to the new frame and new path P
0
. Therefore b
1
V (R
4
),
and hence (u
1
P
1
a
2
Q
2
, Q
3
, . . . , Q
n
) is an a ug menting sequence after the rerouting, contrary
to the choice o f n.
It follows that y
3
6∈ a
1
v
1
. If b
1
V (R
5
), we replace P
1
by Q
1
u
1
P
1
b
1
; then (v
1
P
1
a
2
Q
2
, Q
3
, . . . , Q
n
) is an augmenting sequence that contradicts the choice of n. So it follows
that b
1
u
1
P
1
x
5
. But now (G, Ω) has a gridlet using the paths P
0
, P
k
, Q
1
u
1
P
1
b
1
and a
subpath of R
5
S
5
R
3
C\V (P
0
). This proves (8).
(9) n > 1.
To prove (9) suppose for a contradiction that n = 1. Thus b
1
belongs to the interior of x
4
P x
5
by (1), and a
1
V (P
k
) by (8). But then Q
1
is a T -jump, contrary to (5).
(10) b
1
V (R
3
).
19
To prove (10) we first notice that b
1
V (R
2
R
3
) by (5), (9) and (1). Suppo se for a
contradiction that b
1
V (R
2
). Then a
2
V (R
2
), but b
2
6∈ V (R
1
R
2
R
3
) by (3) and
b
2
6∈ V (P
1
) by (5), a contradiction. This proves (10).
Let P
12
and P
34
be two disjoint subpaths of C, where the first has ends x
1
, x
2
, and the
second has ends x
3
, x
4
. By (8) and (10) the path Q
1
b
1
R
3
x
3
P
34
S
4
is a T -jump disjoint
from R
1
P
12
R
2
, contrary to Lemma 4.4.
We are now ready to prove Theorem 4.1.
Proof of Theorem 4.1. Let (G , ) be a 6-connected society with a leap o f length five. Thus
we may assume that Hyp othesis 4.2 holds for k = 5 . By Lemma 4.8 either (G, Ω) is nearly
rural, in which case the theorem holds, or there exists a leap of length at least four with at
least two exposed vertices. Thus we may assume that there exists a leap of length four with
at least two exposed vertices. Let C be a cycle as in Lemma 4.9. If there is no diverse leap,
then C is a triangle, (G\E(C), Ω) is rurally 4-connected and hence rural by Lemma 4.10,
and the theorem holds. Thus we may assume t hat the cycle C is not a tria ngle, and so by
Lemma 4.11 we may assume that Hypothesis 4.13 f or k = 4 holds. By Lemma 4.14 and the
6-connectivity of G t here is no separation (A, B) as described in Lemma 4.15, and hence by
that lemma there exists an augmenting sequence (Q
1
, Q
2
, . . . , Q
n
). By Lemma 4.16 the path
Q
1
intersects P
2
, and hence Q
1
is disjoint from P
3
, contrary to Lemma 4.16 applied to the
leap (P
0
, P
1
, P
3
, P
4
) of length three and an augmenting sequence (Q
1
, Q
2
, . . . , Q
n
), where Q
1
is the union of Q
1
and a
1
P
2
u
2
or a
1
P
2
v
2
.
5 Societies of bounded depth
Let (G, Ω) be a society. A linear decomposition of (G, Ω) is an enumeration {t
1
, . . . , t
n
} of
V (Ω) where (t
1
, . . . , t
n
) is clockwise, together with a family (X
i
: 1 i n) of subsets of
V (G), with the following properties:
(i)
S
(X
i
: 1 i n) = V (G),
(ii) for 1 i n, t
i
X
i
, and
(iii) for 1 i i
i
′′
n, X
i
X
i
′′
X
i
.
The depth of such a linear decomposition is
max(|X
i
X
i
| : 1 i < i
n),
and the depth of (G, ) is the minimum depth of a linear decomposition of (G, Ω). Theo-
rems (6.1), (7.1) and (8.1) of [9] imply the following.
Theorem 5.1 There exists an integer d such that every 4-conn ected society (G, Ω) either
has a separated doublecross, three crossed paths or a leap of length fi ve, or some planar
truncation of (G, Ω) has depth at most d.
20
In light of Theorems 4.1 and 5.1, in the remainder of the paper we concentrate on societies
of bounded depth. We need a few definitions. Let (G, Ω) be a society, let u
1
, u
2
, . . . , u
4t
be
clockwise in Ω, and let P
1
, P
2
, . . . , P
2t
be disjoint bumps in G such that for i = 1, 2, . . . , 2t
the path P
2i1
has ends u
4i3
and u
4i1
, and the path P
2i
has ends u
4i2
and u
4i
. In those
circumstances we say that (G, Ω) has t disjoint consecutive cross e s .
Now let u
1
, v
1
, w
1
, u
2
, v
2
, w
2
, . . . , u
t
, v
t
, w
t
be clockwise in Ω, let x V (G) {u
1
, v
1
,
w
1
, . . . , u
t
, v
t
, w
t
}, for i = 1, 2, . . . , t let P
i
be a path in G \x with ends u
i
and w
i
and
otherwise disjoint fr om V (Ω), let Q
i
be a path with ends x and v
i
and otherwise disjoint
from V (Ω), and assume that the paths P
i
and Q
i
are pa irwise disjoint, except that the paths
Q
i
meet at x. Let W be the union of all the paths P
i
and Q
i
. We say that W is a windmill
with t vanes, and that the graph P
i
Q
i
is a van e of the windmill.
Finally, let u
1
, u
2
, . . . , u
t
and v
1
, v
2
, . . . , v
t
be vertices of V (Ω) such that for all x
i
{u
i
, v
i
}
the sequence x
1
, x
2
, . . . , x
t
is clockwise in Ω. Let z
1
, z
2
V (G)−{u
1
, v
1
, . . . , u
t
, v
t
} be distinct,
for i = 1, 2, . . . , t let P
i
be a path in G\z
2
with ends z
1
and u
i
and otherwise disjoint from
V (Ω), and let Q
i
be a path in G\z
1
with ends z
2
and v
i
and otherwise disjoint from V (Ω).
Assume that the paths P
i
and Q
j
are disjoint, except that the P
i
share z
1
, the Q
i
share z
2
and P
i
and Q
i
are allowed to intersect. Let F be the union of all the paths P
i
and Q
i
. Then
we say that F is a fan with t blades, and we say t hat P
i
Q
i
is a blade of the fan. The vertices
z
1
and z
2
will be called the hubs of the fan. In Section 8 we prove the following theorem.
Theorem 5.2 Fo r every two integers d and t there exists an integer k such that every 6-
connected k-cosmopolitan society (G, Ω) of de pth at most d contains one of the following:
(1) t disjoint consecutive crosses , or
(2) a windmill with t vanes, or
(3) a fan with t blades.
Unfortunately, windmills a nd fans are nearly rural, and so for our application we need to
improve Theorem 5.2. We need more definitions.
Let x, u
i
, v
i
, w
i
, P
i
, Q
i
be as in the definition of a windmill W with t vanes, let a, b, c, d
V (G) be such that u
1
, v
1
, w
1
, . . . , u
t
, v
t
, w
t
, a, b, c, d is clockwise in Ω, and let (P, Q) be a cross
disjoint from W whose paths have ends in { a, b, c, d}. In those circumstances we say that
W P Q is a wi ndmill with t vanes and a cross .
Now let u
i
, v
i
, P
i
, Q
i
be as in the definition of a fan F with t blades, and let a, b, c, d V (Ω)
be such that all x
i
{u
i
, v
i
} the sequence x
1
, x
2
, . . . , x
t
, a, b, c, d is clockwise in Ω. Let (P, Q)
be a cross disjoint from F whose paths have ends in {a, b, c, d}. In those circumstances we
say that W P Q is a fan with t blades and a cross.
Let z
1
, z
2
, u
i
, v
i
, P
i
, Q
i
be as in the definition of a fan F with t blades, and let a
1
, b
1
, c
1
, a
2
,
b
2
, c
2
V (G) be such that all x
i
{u
i
, v
i
} the sequence x
1
, x
2
, . . . , x
t
, a
1
, b
1
, c
1
, a
2
, b
2
, c
2
is
clockwise in Ω, except that we permit c
1
= a
2
. For i = 1, 2 let L
i
be a path in G \V (F ) with
21
ends a
i
and c
i
and otherwise disjoint from V (Ω), and let S
i
be a path with ends z
i
and b
i
and otherwise disjoint from V (F ) V (Ω). If the paths L
1
, L
2
, S
1
, S
2
are pairwise disjoint,
except possibly for L
1
intersecting L
2
at c
1
= a
2
, then we say that F L
1
L
2
S
1
S
2
is
a fan with t b l ades and two jumps.
Now let u
i
, v
i
, P
i
, Q
i
be as in the definition of a fan F with t+1 blades, and let a, b V (Ω)
be such that all x
i
{u
i
, v
i
} the sequence x
1
, x
2
, . . . , x
t
, a, x
t+1
, b is clockwise in Ω. Let P be
a path in G\V (F ) with ends a and b, and otherwise disjoint from V (F ). We say that F P
is a fan with t blades and a jump. In Section 9 we improve Theorem 5.2 as follows.
Theorem 5.3 Fo r every two integers d and t there exists an integer k such that every 6-
connected k-cosmopolitan society (G , ) of depth at most d is either nearly rural, o r contains
one of the followi ng:
(1) t disjoint consecutive crosses , or
(2) a windmill with t vanes and a cross, or
(3) a fan with t blades and a cross, or
(4) a fan with t blades and a jump, or
(5) a fan with t blades and two j ump s.
For t = 4 each of the above o utcomes gives a turtle, and hence we have the following
immediate corollary.
Corollary 5.4 Fo r every integer d there exists an integer k such that every 6-connected
k-cosmopo l itan s ociety (G, Ω) of depth at most d is either nearly rural, or has a turtle.
The next three sections are devoted to proo fs of Theorems 5.2 and 5.3. The proof of
Theorem 5.2 will be completed in Section 8 and the proo f of Theorem 5.3 will be completed
in Section 9. At that time we will be able to deduce Theorem 1.8.
6 Crosses and goose b umps
In this section we prove that a society (G, Ω) either satisfies Theorem 5.2, or it has many
disjoint bumps. If X is a set and is a cyclic permutation, we define \X t o be |(V (Ω)
X). Let P
1
, P
2
, . . . , P
k
be a set of pairwise disjoint bumps in (G, Ω), where P
i
has ends u
i
and v
i
and u
1
, v
1
, u
2
, v
2
, . . . , u
k
, v
k
is clockwise in Ω. In those circumstances we say that
P
1
, P
2
, . . . , P
k
is a goose b ump in (G, Ω) of strength k.
Lemma 6.1 Let b, d and t be positive integers, and let (G, Ω) be a society of depth at most
d. T hen either (G, Ω) has a goose bump of strength b, or there is a set X V (G) of size at
most (b 1)d such that the society (G\X, \X) has no bump.
22
Proof. Let (t
1
, t
2
, . . . , t
n
) and (X
1
, X
2
, . . . , X
n
) be a linear decomposition of (G, Ω) of depth
at most d, a nd for i = 1 , 2, . . . , n 1 let Y
i
= X
i
X
i+1
. If P is a bump in (G, Ω), then the
axioms of a linear decomposition imply that
I
P
:= {i {1, 2, . . . , n 1} : Y
i
V (P ) 6= ∅}
is a nonempty subinterval of {1, 2, . . . , n 1}. It follows that either there exist bumps
P
1
, P
2
, . . . , P
b
such that I
P
1
, I
P
2
, . . . , I
P
b
are pairwise disjoint, or there exists a set I
{1, 2, . . . , n 1} of size a t most b 1 such that I I
P
6= for every bump P . In the former
case P
1
, P
2
, . . . , P
b
is a desired goose bump, and in the latter case the set X :=
S
iI
Y
i
is as
desired.
The proof of the f ollowing lemma is similar and is omitted.
Lemma 6.2 Let t and d be positive integers, and let (G, Ω) be a society of depth at most d.
Then either (G, Ω) has t disjoint consecutive crosse s , or there is a set X V (G) of size at
most (t 1)d such that the soci ety (G\X, \X) is cross-free.
Lemma 6.3 Let d, b, t be positive integers, let k (b 1)d + (t 1 )
(b1)d
2
+ 1 and let
(G, Ω) be a 3-connec ted society of depth at mos t d such that at least k vertices in V (Ω) have
at least two neighbors in V (G). Then (G, Ω) has either a fan with t blades, or a goose bump
of strength b.
Proof. By Lemma 6.1 we may assume that t here exists a set X V (G ) of size at most
(b 1)d such that (G\X, \X) has no bump. There are at least (t 1)
(b1)d
2
+ 1 vertices
in V (Ω) X with at least two neighbors in V (G). Let v be one such vertex, and let H
be the component of G\X containing v. Since (G\X, \X) has no bumps it f ollows that
V (H) V (Ω) = {v} . By the fact that v has at least two neighbors in G (if V (H) = {v}) or
the 3-connectivity of (G, Ω) (if V (H) 6= {v}) it follows that H has at least two neighbors in
X. Thus there exist distinct vertices z
1
, z
2
such that for at least t vertices of v V (Ω) X
the component of G\X containing v has z
1
and z
2
as neighbors. It follows that (G, Ω) has
a fan with t blades, as desired.
7 Intrusions, invasions and wars
Let be a cyclic permutation. A base in is a pair (X, Y ) of subsets of V (Ω) such that
|X Y | = 2, X Y = V (Ω) and fo r distinct elements x
1
, x
2
X and y
1
, y
2
Y the sequence
(x
1
, y
1
, x
2
, y
2
) is not clockwise. Now let (G, Ω) be a society. A separation (A, B) o f G is
called an i ntrusion in (G, Ω) if there exists a base (X, Y ) in such that X A, Y B and
there exist disjoint paths (P
v
)
vAB
, each with one end in X, the other end in Y and with
23
v V (P
v
). The intrusion (A, B) is minimal if there is no intrusion (A
, B
) of order |A B|
with base (X, Y ) such that A
is a proper subset of A. The pat hs P
v
will be called longitudes
for the intrusion (A, B). We say that (A, B) is based at (X, Y ), and that (X,