Four-Terminal Reducibility
and Projective-Planar
Wye-Delta-Wye-Reducible Graphs
Dan Archdeacon
Dept. of Math. and Stat.
University of Vermont
Burlington, VT 05405 USA
Charles J. Colbourn
Dept. of Computer Science
University of Vermont
Burlington, VT 05405 USA
Isidoro Gitler
Departamento de Matematicas
Centro de Investigacion
y Estudios Avanzados del IPN
Apartado Postal 14{740
Mexico, D.F. 07000 MEXICO
J. Scott Provan
Operations Research
University of North Carolina
Chapel Hill, NC 27599 USA
dan.archdeacon@uvm.edu
colbourn@emba.uvm.edu
scott provan@unc.edu
igitler@math.cinvestav.mx
Revised May 1, 1998
Abstract
A graph is Y Y -reducible if it can be reduced to a vertex by a
sequence of series-parallel reductions and Y Y -transformations. Terminals are distinguished vertices which cannot be deleted by reductions and transformations. In this paper we show that four-terminal
planar graphs are Y Y -reducible when at least three of the vertices
lie on the same face. Using this result we characterize Y Y -reducible
projective-planar graphs. We also consider terminals in projectiveplanar graphs, and establish that graphs of crossing-number one are
Y Y -reducible.
1
1 Introduction
A variety of results in graph theory show that a class of graphs can be reduced
to a canonical form by repeatedly applying certain operations. Equivalently,
this shows that the repeated application of the inverse operations can generate all graphs in a class. For example, Tutte (Theorem IV58 in [28]) showed
that every 3-connected graph can be generated from a wheel by a sequence
of vertex splittings and edge additions. These types of characterizations are
useful in inductive proofs.
One of the more important sets of reductions is the series-parallel reductions. These apply to graphs which may include loops, an edge whose two
end-vertices are identical, and parallel edges, two edges with the same pair
of end-vertices. The four reductions are:
R0 ) Loop reduction: Delete a loop.
R1 ) Degree-one reduction: Delete a degree-one vertex and its incident edge.
R2 ) Series reduction: Delete a degree-two vertex y and its two incident
edges xy and yz, and add in a new edge xz.
R3 ) Parallel reduction: Delete one of a pair of parallel edges.
Each of these reductions decreases the number of edges in a graph. A
connected graph is series-parallel reducible if it can be reduced to a single
vertex by a sequence of these operations. Some authors do not allow degreeone reductions in their series-parallel reductions. This restriction changes the
class of series-parallel reducible graphs (or more commonly the set of reduced
graphs is changed so that the reducible graphs remain the same), but the two
classes are equivalent if the graph is 2-connected. Similarly, we could expand
our reductions to allow deletion of isolated vertices, which would extend the
results to disconnected graphs.
Two other transformations of graphs are important. A wye (Y ) is a vertex
of degree three. A delta () is a cycle of length three. The transformations
are:
Y ) Wye-delta transformation. Delete a wye w and its three incident edges
wx, wy, wz, and add in a delta xyz.
2
Y ) Delta-wye transformation. Delete the edges of a delta xyz, and add in
a new vertex w and new edges wx, wy, and wz.
Two graphs are Y Y -equivalent if one can be changed into the other by
a sequence of Y - and Y -transformations. A connected graph is Y Y reducible if it can be reduced to a single vertex by a sequence of seriesparallel reductions and Y or Y transformations. Not every graph is
Y Y -reducible: for example, a simple graph of minimal degree four and
girth four admits no reductions or transformations.
No characterization of Y Y -reducible graphs is known, but some partial
results are interesting. For example, Akers [1] and Lehman [17] conjectured
that every planar graph is Y Y -reducible. This was rst proved by Epifanov
[9] in 1966 (see also [15]). Simpler proofs are given by Feo and Provan [11],
and Truemper [21]. Gitler [13] showed that graphs with no K5 minor are
Y Y -reducible, and also that graphs with no K3 3 minor are Y Y -reducible.
Contracting an edge e = uv consists of deleting e and identifying its two
endpoints u = v to make a single vertex. A minor of G is a graph formed
by a sequence of edge deletions, edge contractions, and deletion of isolated
vertices.
A class of graphs is minor closed if whenever a graph G is in the class, any
minor H of G is also in the class. Wagner conjectured that any minor-closed
class of graphs could be characterized by a nite set of forbidden minors.
In other words, the set of minor-minimal graphs not in the class must be
nite. For example, the class of series-parallel graphs is minor-closed and
is characterized by the exclusion of a K4 minor. Wagner's conjecture was
proved by Robertson and Seymour [20] in a remarkable sequence of papers.
Truemper [21] showed that the class of Y Y -reducible graphs is minor
closed (he assumed that the minor was 2-connected, since he did not allow
degree-one reductions). From the above it follows that there are a nite
number of minor-minimal graphs which are not Y Y -reducible. One such
graph is K6 (for a proof see [27]). It follows that any graph which is Y Y equivalent to K6 is also minor-minimal non-Y Y -reducible. There are seven
such graphs. They are called the Petersen family since the Petersen graph is
among them.
A variation on Y Y -reducibility is to forbid reductions on some distinguished vertices. Specically, let T V (G) be a set of terminals. A terminal
cannot be deleted in a degree-one or series reduction, nor can it be deleted in
;
3
a Y -transformation. If a graph with terminals can be reduced to eliminate
all non-terminal vertices, then we say it is (terminal) Y Y -reducible.
Akers also conjectured [1] that any 3-terminal planar graph is Y Y reducible. This conjecture was proved by Gitler [13, 14]. On the other
hand, there are examples of innite families of 4-terminal planar non-Y Y reducible graphs [1, 11, 13].
Combining the previous two concepts, we can have a minor H of a graph
G with terminals. Specically, a terminal minor is formed using the same
three minor operations as above, except that we forbid contracting an edge
joining two terminals and deleting an isolated terminal. It follows that H
has the same number of terminals as G.
The rst result in this paper proves that any terminal minor of a Y Y reducible graph is itself Y Y -reducible. Using this result, we prove that any
4-terminal planar graph is Y Y -reducible if at least three of the vertices are
on the boundary of a common face. These results are presented in Section
2. An application of these results allows us to characterize projective-planar
Y Y -reducible graphs in Section 3. We also get some results on reducing
projective-planar graphs with terminals, and show that graphs with crossing
number one are Y Y -reducible. We close in Section 4 with some conclusions
and directions for future research.
2 Four-Terminal Reducibility
Truemper [21] mentioned that any 2-connected minor of a 2-connected Y Y reducible graph was itself Y Y -reducible. Gitler [13] extended this result to
graphs with terminals. We now give a version with terminals but without
connectivity constraints.
Theorem 2.1 The Minor Theorem: Suppose that H is a terminalminor of G. If G is terminal Y Y -reducible, then H is terminal Y Y reducible.
Proof: Suppose that G is formed from G by one of the four reductions
0
or two transformations. We will construct a terminal minor H of G by
applying a corresponding operation on H . The result will then follow by
induction on the number of operations needed to reduce G.
0
4
0
Partition the edges of G into three parts depending on their fate when
forming H : D is the set of deleted edges, C is the set of contracted edges,
and M is the set of edges corresponding to those in H .
If G is formed from G by a loop reduction of an edge in C or D, then
set H = H . If the reduced loop is in M , then form H by the corresponding
loop reduction on H . Then H is a terminal minor of G . Similarly, if G is
formed by a degree-one reduction on an edge in D or C , then set H = H . If
the edge is in M , then the degree-one vertex in H cannot be a terminal and
we form H by the corresponding degree-one reduction.
If G is formed by a series reduction, then set H = H if either one of the
two incident edges is in C or if both edges are in D. Hence either zero or one
edge is in D and the corresponding vertex cannot be a terminal of H . Form
H by the corresponding series or degree-one reduction respectively. If G is
a parallel reduction of G, then set H = H unless both edges are in M , in
which case H is formed by the corresponding parallel reduction on H .
Suppose that G is formed by a Y -transformation of G. If any edge in
the is in D, then H = H is still a minor of G . If at least two edges are
in C , then the corresponds to a single vertex in H . Form H from H by
reducing any loops on this vertex. We conclude that either zero or one edge
is in C and the others are in M . Now form H by a Y -transformation or
parallel reduction respectively on H .
Finally, suppose that G is a Y -transformation of G. If any edges in the
Y are in C , then H = H is still a minor of G . The same holds if all three
edges are in D. If exactly one or two edges of the Y are in D, then the central
vertex corresponds to a degree-one or degree-two non-terminal vertex of H .
Form H by a degree-one or series reduction respectively. Finally, if all three
edges are in M , then form H by the corresponding Y -transformation. In
all cases the h so formed is a terminal minor of G as desired.
0
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We next turn our attention to reducing planar graphs with terminals.
Recall that every 3-terminal planar graph is Y Y -reducible [13, 14], but
there are innitely many 4-terminal planar graphs which are not Y Y reducible [1, 11, 13]. The remainder of this section is devoted to proving
that 2-connected 4-terminal planar graphs are reducible if at least three of
the terminals lie on a common face.
Our basic approach is as follows. We rst embed the 4-terminal planar
graph with at least three terminals on a common face into a planar grid so
5
that the three grid terminals lie on a common grid square and the fourth
terminal lies either on the same square or on a corner. We then show how
to reduce these planar grids. By the Minor Theorem 2.1 our graphs, which
are minors of these grids, are also reducible.
Let P denote a path with q edges. The p q grid is the Cartesian product
P P . We say that a graph G embeds in a grid if there exists a subgraph
H of the grid and a sequence of series reductions which takes H to G. In
other words, the vertices of G are vertices of the grid and the edges of G are
internally-vertex-disjoint paths in the grid.
Valiant [25] showed that a graph G embeds in a grid if and only if it is
planar of maximum degree at most four. The method of proof is important
to this paper. Specically, he showed that the embedding can be constructed
by adding in a vertex at a time. Each vertex and its incident edges are in
the unbounded face of the preceding embedding. Moreover, be showed that
each vertex increases the width p and breadth q of the grid by at most three,
hence there is an upper bound on the area A = pq (3n)2.
q
p
q
Lemma 2.2 Let G be a 2-connected 4-terminal plane graph with exactly
three terminals on a common face. Then there is an embedding of a minor G of G in a grid such that three corresponding grid terminals lie on a
common grid square and the fourth terminal is a corner of the grid.
0
Proof: Label the terminals of G as t1, t2, t3, and t4 where the rst three
lie on a common face F . Embed G so that t4 lies on the unbounded face.
Form a minor G by splitting each vertex of G in the plane so that a) the
degree of each vertex in G is at most three, and b) the face corresponding
to F has three degree-2 vertices which contract to t1, t2, t3. Form G from
G by adding in three edges pairwise connecting these corresponding degree
two vertices. Then G is planar of maximum degree four.
We show that G is isomorphic to a subgraph of a grid using a method
exactly the same as Valiant's. Specically, start by embedding the face F
in a square of a grid. Note that the three grid terminals lie on a common
grid square. Now add one non-terminal vertex at a time as in Valiant's
proof. Finally, add in the fourth terminal. By expanding the grid the fourth
terminal can be placed at a corner as claimed.
0
0
00
0
00
00
0
For technical reasons we also need the following similar lemma.
6
Lemma 2.3 Let G be a 2-connected 4-terminal plane graph with all four
terminals on a common face. Then there is an embedding of a minor G of
G in a grid such that all four corresponding grid terminals lie on a common
grid square.
0
Proof: Let G be a planar graph with four terminals lying on a common face
F . As in the preceding lemma, form G by splitting vertices of G until we
have a graph of maximum degree 3 and a face F containing four degree-two
vertices which contract to the four terminals. Again form G by adding in
four edges making a quadrilateral face F containing all four terminals. Now
embed F on a grid and inductively add vertices as in the preceding lemma.
0
0
00
00
00
The result is the embedding claimed.
By the preceding two lemmas and the Minor Theorem 2.1, the problem
of reducing 2-connected 4-terminal planar graphs is equivalent to reducing 4terminal planar grids. However, it is already known how to reduce 4-terminal
planar grids [13]. Specically, we cite the following.
Proposition 2.4 Let G be a 4-terminal grid with at least three terminals on
a common face and the fourth terminal either on the common face or on a
corner. Then G is Y Y -reducible.
The proof of this proposition given in [13] uses the Corner Algorithm.
The basic idea of the Corner Algorithm is to repeatedly reduce a modied
p q grid to a modied p (q ? 1) grid, then to reduce a modied p 2 grid
to a modied (p ? 1) 2 grid.
There are several other reduction algorithms given in 2.4 and [21]. Neither
of these were concerned with terminals. However, an examination of the
algorithms shows that they avoid reductions on certain vertices. Thus we
could instead use these algorithms for a dierent proof of Proposition 2.4.
Proposition 2.4 and the Minor Theorem 2.1 immediately give the following.
Theorem 2.5 A 4-terminal planar graph with at least three vertices incident
with a common face is Y Y -reducible.
7
3 Nonplanar Graphs
Recall that every planar graph is Y Y -reducible. In this section we extend this result by classifying all projective-planar Y Y -reducible graphs.
The proof hinges on an intriguing connection between the reducibility and
topological properties of the embedded graphs. As a corollary we see that
every graph of crossing number one is reducible. Our results also establish
reducibility for some terminal projective-planar graphs.
The projective plane is the surface formed from the sphere by identifying
antipodal points. Equivalently, we can form it from a disk by identifying
only antipodal points on the boundary. We denote the projective plane by
P 2.
Let G be a graph embedded on the projective plane. The width of the
embedding is the minimum jG \ C j over all non-contractible cycles C . The
width is also known as the representativity of the embedding. It is important
to note three facts about the width of an embedding: a) having width at most
k is closed under minors, b) a Y or Y transformation on a triangular face
does not change the width, and c) the width w of a projective-planar graph
may depend on the embedding, but bw=2c is constant. See [19] for a proof of
the rst two and other facts about the width, and see [12] for a proof of c).
We also refer the reader to Barnette [3] and independently Vitray [19, 26] for
a proof of the following.
Proposition 3.1 There are exactly 6 minor-minimal projective-planar graphs
with embeddings of width at least 3.
These six graphs all belong to the Petersen family. (One member of
the Petersen family, K4 4 ? K2, does not embed in the projective plane. A
second embeds in two non-isomorphic ways, so the six graphs make seven
maps.) Since the Petersen family is not Y Y -reducible, it follows that any
projective-planar graph with width at least 3 is not Y Y -reducible. We turn
our attention to projective-planar graphs of width at most two.
Proposition 3.2 One-terminal projective-planar graphs of width at most
two are Y Y reducible.
Proof: Suppose that G is a projective-planar graph of width at most two
and with at most one terminal. If the width is 0 or 1, then the graph is
planar and hence reducible. So we can assume that the width of G is two.
;
8
Let C be a non-contractible cycle of with jC \ Gj = 2. We suppose
without loss of generality that the intersections occur at two vertices, one of
them v. The curve C determines a bipartition of the edges of G incident with
v dened by whether the incident edge is locally on the left or right side of C .
Replace v in G with two new vertices, v and v . Similarly replace the edges
incident with v on the left and right sides of C with edges incident with v
and v respectively. Call the resulting graph G . Distinguish three terminals
in G : v , v , and u where u is the vertex corresponding to the terminal of G.
There is a natural embedding of G in P 2 with width one, and hence G
also embeds in the plane. Any 3-terminal planar graph is reducible [13, 14],
so there is a sequence of Y Y -transformations and series-parallel reductions which eliminates all non-terminal vertices of G . The corresponding
sequence eliminates all non-terminal vertices of G except v. But v can now
be eliminated by a sequence of parallel reductions followed by a degree-one
reduction.
`
r
`
0
r
0
`
r
0
0
0
Combining the previous two propositions, we get a characterization of
Y Y -reducible projective-planar graphs.
Theorem 3.3 A projective-planar graph G is Y Y -reducible if and only
if its projective-plane embedding has width at least three. Equivalently, G
is Y Y -reducible if and only if it has no minor in the Petersen family.
Moreover the above two statements hold if G has a single terminal.
Theorem 3.3 cannot be extended to graphs with two or more terminals.
Consider the graph K6 ? e with two terminals at the degree four vertices.
This graph embeds in the projective plane with width two. It is not reducible,
since any sequence of transformations and reductions would correspond to
a reduction of K6. Hence, in general, two-terminal projective-planar graphs
are not reducible. The following proposition covers some cases where twoterminal reducibility is possible.
Proposition 3.4 Let G be a projective-planar graph of width two and with
terminals a; b. Suppose that there is an essential cycle C which intersects G
in exactly two points and at least one of a; b lie on the boundary of a face
containing a portion of C . Then G is Y Y -reducible.
9
Proof: The proof is similar to that of Proposition 3.2. Again, we suppose
that v 2 C \ G, and split v into two vertices v and v with edge incidences
`
r
determined by the left and right sides of C at v. The resulting graph is planar
with four terminals, a, b, v and v . If a is the vertex on the boundary of a
face with a portion of C , then fa; v ; v g all lie on a common face of a planar
embedding. The reducibility follows by Theorem 2.5
`
r
`
r
The crossing number of a graph is the minimum number of pairwise intersections of edges over all drawings of the graph in the plane. Any graph
with crossing number one embeds in the projective plane with width two
(see, e.g., [18]). Hence we easily get the following corollary of Proposition
3.2.
Corollary 3.5 Any graph with crossing-number one and at most one terminal is Y Y -reducible.
Corollary 3.5 also follows immediately from the reducability of 3-terminal
planar graphs: merely delete one of the crossing edges and mark its two
ends as two additional terminals. Finally, note that the Petersen graph has
crossing number two and is not Y Y -reducible, so Corollary 3.5 is tight.
4 Conclusions
The concept of Y Y -reducibility has occurred naturally in several recent
papers [4, 6, 16, 22, 23]. In this paper we have shown that 4-terminal planar
graphs are reducible in the case that at least three terminals lie on a common
face. Using this, we showed that projective-planar graphs are reducible if and
only if they have width at most two. Moreover, we can x one terminal in
the projective-planar graph. One-terminal graphs with crossing-number one
are included in this class.
The foremost related open problem is the following.
Problem 4.1 Find the minor-minimal non-Y Y -reducible graphs.
To quote Truemper ([24], p.100) \One might conjecture, and would not
be the rst person to do so, that the [Petersen family] constitutes all minimal
nonreducible graphs. The conjecture is appealing but false." One such graph
10
is K5 5 ? M where M is a perfect matching. It is 4-regular of girth 4, so no
reductions are possible. On the other hand, the symmetry makes it easy to
verify that any minor is Y Y -reducible.
Archdeacon [2] has found a forbidden topological-subgraph characterization of projective-planar graphs. An analysis of this set reveals that there are
exactly 12 Y -minor-minimal members of this set (one is disconnected). An
examination of these 12 graphs reveals that 11 of them are Y Y -reducible.
The twelfth, K4 4 ? K2, is a member of the Petersen family. Thus it appears
that other minimal non-Y Y -reducible graphs are fairly large.
There has been some work done on the case that either the Y or Y transformation is not available. A summary of these results can be found in
El-Mallah and Colbourn [8].
The complexity of reducing a planar graph with n vertices is still open.
The fastest known algorithms [10, 11] have complexity O(n2) and they seem
dicult to improve. On the other hand, for the special case of an r r
planar grid pthe algorithm in [21] and the corner algorithms in [13, 14] run
and analysis of
in time O(r r). It is possible that a careful implementation
p
the grid-embedding schemes can lead to an O(n n)-time algorithm for the
general planar case. It would be interesting to obtain a near-linear algorithm
for the grid, since many applications are either naturally stated for grids, or
their most important cases are those restricted
p to grids. However, it may
very well be that reducing planar grids is (n n).
Multi-terminal algorithms provide a method for obtaining a substantial
reduction in the size of an arbitrary graph with terminals. In solving multicommodity ow problems, for example, we get a graph in which the commodity ows can be found either directly or by solving a reasonably small
linear program (see Feo [10]).
In the same vein, another interesting application is to the computation
of source-to-sink connectedness reliability in a planar graph with randomly
independently failing edges. In [4] this is done using Lehman's Y Y approximation [17]. Lehman's work may apply to the multi-terminal case as well,
particularly for the case of three terminals. This further suggests a good
approximation heuristic for the problem of computing the probability that
a given set K of vertices can be connected pairwise by operating paths in
the network (the K ? reliability problem), by using known decomposition
methods [5].
;
;
11
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