Min-max relations for odd cycles in planar graphs

Abstract

Let m(G) be the maximum number of vertex-disjoint odd cycles of a graph G and
t(G) the minimum number of vertices whose removal makes G bipartite. We show
that t(G)<=6m(G) if G is planar. This improves the previous bound t(G)<=10m(G)
by Fiorini, Hardy, Reed and Vetta [Math. Program. Ser. B 110 (2007), 71-91].

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arXiv:1108.4281v1 [math.CO] 22 Aug 2011
Min-max relations for odd cycles in planar graphs∗
Daniel Kr´ al’†
Jean-S´ ebastien Sereni‡
Ladislav Stacho§
Abstract
Let ν(G) be the maximum number of vertex-disjoint odd cycles of a
graph G and τ(G) the minimum number of vertices whose removal makes
G bipartite. We show that τ(G) ≤ 6ν(G) if G is planar. This improves
the previous bound τ(G) ≤ 10ν(G) by Fiorini, Hardy, Reed and Vetta
[Math. Program. Ser. B 110 (2007), 71–91].
1Introduction
Packing problems are among the most important problems in combinatorial op-
timization. In this paper, we focus on the problem of packing odd cycles in
graphs. If G is a graph, let ν(G) be the size of a maximum collection (packing) of
vertex-disjoint odd cycles of G, and τ(G) the size of a minimum set S of vertices
(transversal) such that each odd cycle of G contains at least one vertex of S
(which is is equivalent to G \ S being bipartite). Clearly, ν(G) ≤ τ(G).
One of the most studied questions on packing problems is whether the size
of a maximum packing can be bounded by a function of the size of a minimum
transversal. If this is the case, the problem is said to have the Erd˝ os-P´ osa prop-
erty. The name is due to the result of Erd˝ os and P´ osa [3] who established this
property for packing (general) cycles in graphs. For general graphs, the packing
problem for odd cycles does not have the Erd˝ os-P´ osa property. Reed [9] gave an
∗This research was done in the framework of the Czech-French project PHC Barrande
24444XD (the Czech side reference: MEB021115).
†Department of Applied Mathematics and Institute for Theoretical Computer Science (ITI),
Faculty of Mathematics and Physics, Charles University, Malostransk´ e n´ amˇ est´ ı 25, 118 00
Prague 1, Czech Republic. E-mail: kral@kam.mff.cuni.cz. Institute for Theoretical computer
science is supported as project 1M0545 by Czech Ministry of Education.
‡CNRS (LIAFA, Universit´ e Denis Diderot), Paris, France, and Department of Applied Math-
ematics (KAM), Faculty of Mathematics and Physics, Charles University, Prague, Czech Re-
public. E-mail: sereni@kam.mff.cuni.cz.
§Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby,
B.C., V5A 1S6, Canada. E-mail: lstacho@sfu.ca. This author’s stay at LIAFA was partially
supported by the French Agence nationale de la recherche under the reference ANR 10 JCJC
0204 01.
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example of a projective planar graph G with τ(G) arbitrary large and no two
vertex-disjoint odd cycles, i.e., ν(G) = 1. These graphs are called Escher walls.
In fact, they play a key role for the problem. The main result from [9] asserts
that the problem of packing odd cycles in a minor-closed family of graphs has
the Erd˝ os-P´ osa property if and only if the family avoids Escher walls of arbitrary
height.
Since the class of planar graphs avoids all Escher walls, it follows that there
exists a function f such that τ(G) ≤ f(ν(G)) if G is planar. However, the function
given by the methods from [9] is enormous since the proof is based on the graph
minor machinery. So, it is natural to search for better estimates for particular
graph classes. In [4], Fiorini, Hardy, Reed and Vetta showed that τ(G) ≤ 10ν(G)
for planar graphs. The purpose of this article is to further improve this estimate
to τ(G) ≤ 6ν(G) (Theorem 9). These results also hold in a more general setting
where edges are assigned parities. Since our proof is constructive and all its steps
can be efficiently performed, we also obtain the existence of a polynomial-time
6-approximation algorithm for the odd cycle packing problem in planar graphs
(Corollary 10) which improves the 11-approximation algorithm given in [4]. The
problem is known [5] to be NP-hard.
We do not believe the multiplicative constant in Theorem 9 is optimal. In
fact, we are not aware of an example showing it exceeds two. This multiplicative
factor is known [8] to be true, i.e., τ(G) ≤ 2ν(G), if G is highly connected (the
connectivity depends on ν(G)). The optimal constant is however known for the
edge version of the problem in the planar case. Similarly to the vertex case,
there is no function bounding the edge transversal τe(G) in terms of the size
νe(G) of a maximum collection of edge-disjoint odd cycles for general graphs G.
However, for planar graphs, such a function exists [1], and the optimal estimate
τe(G) ≤ 2νe(G) was proven in [6]; its more compact proof can be found in [4].
Another related problem is a conjecture of Tuza which asserts that the min-
imum size τt(G) of a set of edges intersecting every triangle is at most twice
the maximum number νt(G) of edge-disjoint triangles. The conjecture is known
to be true for planar graphs [12] and it is also known that two of its fractional
relaxations hold [7].
2 Notation
We work in the more general setting of signed graphs. In this setting, each edge
is assigned a parity, i.e., it is odd or even. A cycle is said to be odd if it contains
an odd number of odd edges and it is even otherwise. A face of a plane signed
graph is odd if its boundary contains an odd number of odd edges (bridges are
counted twice); it is even otherwise. It is easy to see that the boundary of an odd
face must contain an odd cycle. The property whether a cycle is odd or even is
referred to as its parity.
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Since we exclusively focus on odd cycles in signed graphs, we call a set S of
vertices of a signed graph G a transversal of G if G \ S has no odd cycle. A
collection C of cycles is a packing if the cycles in C are vertex-disjoint. The two
parameters central to our study are τ(G) which stands for the minimum size of a
transversal of G and ν(G) which is the maximum number of odd cycles forming
a packing in G. In case that all edges of a signed graph G are odd, a cycle in G
is odd if and only if its length is odd, so the definitions coincide with those given
in Section 1.
Signed graphs we consider in this paper are always assumed to be simple.
This does not decrease the generality of our results: if a signed graph G contains
parallel edges, we can subdivide each parallel edge in such a way that one of the
new edges has the same parity as the original edge and the other one is even. It is
not hard to observe that this operation affects neither τ(G) nor ν(G) (considering
pairs of parallel edges with different parities as odd cycles of length two).
2.1
T-joins and T-cuts
A key ingredient for our proof is the notion of a T-join from combinatorial op-
timization. The algorithm for finding a minimum-size T-join forms the core of
the algorithm for solving the max-cut problem for planar graphs. The planar
max-cut problem is dual to finding the minimum set of edges whose removal
bipartizes a given planar graph, which is the quantity τedefined earlier. So, it
is not surprising that the proof of an earlier bound in [4] as well as our proof
use this notion. In fact, the arguments we use in Section 4 can be viewed as an
extension of those given in [4, Subsection 4.3].
We now present the notion and results we later need. The reader can find
a more detailed introduction in monographs on combinatorial optimization, e.g.,
[2, 10]. A T-join in a connected graph G with a distinguished even-size set T of
its vertices is a subgraph J such that the odd degree vertices of J are precisely
the vertices of T. The size of a T-join J is the number of edges it contains
and it is denoted by |J|. The problem of finding a minimum-size T-join can
be reduced to the weighted perfect matching problem on complete graphs which
is well-understood and efficiently solvable. The reduction is as follows: form a
complete graph with vertex set T and assign each edge tt′the length dG(t,t′) of
the shortest path between t and t′in G. For a minimum weight perfect matching
in the auxiliary graph, define a subgraph J to be the union of the shortest paths
corresponding to the edges of the matching (it can be shown that the paths are
edge-disjoint if the perfect matching has minimum weight). So, J forms a T-join
in G which is minimum.
It is well-known that the problem of finding a minimum weight perfect match-
ing can be formulated as a linear program. Considering its dual, we obtain the
following optimization problem with variables yv, v ∈ T, and yS, S ∈ O(T),
where O(T) stands for the set of all odd-size subsets of T with at least three
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elements.
yS
≥ 0
≤ dG(t,t′) for every pair t,t′∈ T
for S ∈ O(T)
yt+ yt′ +?
min?
The duality of linear programming implies that the optimum value of the linear
program (Y) is equal to the minimum size of a T-join in a graph G. A solution
of (Y) is called laminar if yS> 0 and yS′ > 0 implies that either S and S′are
disjoint or one is a subset of the other. It is well-known that the linear program
(Y) has always an optimum laminar solution. Moreover, since the weights of the
edges satisfy the triangle inequality, there is an optimum laminar solution of (Y)
with all variables being non-negative. In case that the weights of the edges in
the auxiliary graph are even (which happens, e.g., when G is bipartite), there
always exists an optimum integral solution of (Y) which is also non-negative and
laminar. We summarize these observations in the next proposition.
S∈O(T),|S∩{t,t′}|=1yS
t∈Tyt+?
S∈O(T)yS
(Y)
Proposition 1. Let G be a connected graph with a distinguished even-size set T
of its vertices. The value of the optimum solution of the linear program (Y) is
equal to the minimum size of a T-join of G. Moreover, there exists an optimum
solution of (Y) that is non-negative and laminar, and if G is bipartite, there exists
an optimum solution that is non-negative, laminar and integral.
By the duality of linear programming, if a T-join J in a graph G has the size
equal to a value of a solution y of (Y), then J is a minimum size T-join and y is
an optimum solution of (Y). We use this fact in the proof of Lemma 5 where we
keep such a pair through the induction, so the T-joins we consider are optimum.
A combinatorial structure dual to a T-join is a T-cut: a T-cut is an edge cut
that splits the graph into two parts each containing an odd number of vertices
of T. It is known [11] that if G is bipartite, then the size of a minimum T-join is
equal to the maximum number of edge-disjoint T-cuts of G. More insight in the
structure of optimum T-joins and solutions of (Y) can be derived by analyzing
specific procedures for obtaining them. We state one such condition based on the
blossom algorithm for the (weighted) perfect matching problem. To do so, we
define a vertex v of a graph G to be k-close to a set S ∈ O(T) with respect to a
solution y of (Y) if
min
t∈S



dG(t,v) − yt−
?
t∈S′?=S
S′∈O(S)
yS′



= k
where dG(t,v) is the distance between t and v in G. A solution of (Y) is a moat
solution if
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• y is non-negative, integral and laminar,
• for every inclusion-wise minimal set S ∈ O(T) with yS > 0, there exists
an ordering t1,...,tsof vertices of S such that yti+ yti+1= dG(ti,ti+1) for
every i = 1,...,s (indices modulo s), and
•?
t∈TCt∪?
distance k and k + 1 from t in G for k = 0,...,yt− 1, and CS, S ∈ O(T),
is the collection of yST-cuts formed by the edges joining pairs of vertices
that are k-close and (k + 1)-close to S in G for k = 0,...,yS− 1.
S∈O(T)CSis a collection of edge-disjoint T-cuts where Ct, t ∈ T,
is the collection of ytT-cuts formed by the edges joining pairs of vertices at
A moat solution always exists if G is bipartite. Let us state this as a separate
proposition.
Proposition 2. Let G be a connected bipartite graph with a distinguished even-
size set T of its vertices. There exists an optimum solution of (Y) that is a moat
solution.
Observe that every optimum T-join intersects every T-cut in the collection
t∈TCt∪?
T-cut and the size of an optimum T-join is equal to the number of T-cuts in the
collection.
?
S∈O(T)CSfrom the definition of a moat solution and this intersection
is formed by a single edge: this follows from that every T-join intersects every
3 Faces in clouds
In this section, we analyze sets of odd faces that are “connected” in a considered
plane signed graph. Formally, we define the vertex-face incidence graph VF(G)
of a plane signed graph G to be the bipartite graph with vertex set formed by
vertices and faces of G such that the vertex of VF(G) associated with a face f
of G is adjacent to the vertices of G incident with f. The subgraph of VF(G)
induced by the vertices of G and the odd faces of G is denoted by VFodd(G).
Finally, a cloud is a set of all odd faces in the same component of the graph
VFodd(G).
We start with the following lemma.
Lemma 3. Let G be a plane 3-connected signed graph and R a cloud of G.
There exists a set F of vertex-disjoint faces of R and a set of vertices U with the
following properties:
P1 each face of R is incident with at least one vertex of U,
P2 each vertex of U is incident with a face of F, and
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