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Revista de la Unión Matemática Argentina
versão impressa ISSN 0041-6932
Rev. Unión Mat. Argent. vol.50 no.1 Bahía Blanca jun. 2009
Exponential families of minimally non-coordinated graphs
Francisco Soulignac* and Gabriel Sueiro
Abstract. A graph G is coordinated if, for every induced subgraph H of G, the minimum number of colors that can be assigned to the cliques of H in such a way that no two cliques with non-empty intersection receive the same color is equal to the maximum number of cliques of H with a common vertex. In a previous work, coordinated graphs were characterized by minimal forbidden induced subgraphs within some classes of graphs. In this note, we present families of minimally non-coordinated graphs whose cardinality grows exponentially on the number of vertices and edges. Furthermore, we describe some ideas to generate similar families. Based on these results, it seems difficult to find a general characterization of coordinated graphs by minimal forbidden induced subgraphs.
* Partially supported by UBACyT Grant X184, Argentina and CNPq under PROSUL project Proc. 490333/2004-4, Brazil.
Let be a graph, with vertex set
and edge set
. Denote by
the complement of
. Given two graphs
and
we say that
contains
if
is isomorphic to an induced subgraph of
. If
, we denote by
the subgraph of
induced by
.
A complete set or just a complete of is a subset of vertices pairwise adjacent. A clique is a complete set not properly contained in any other. We may also use the term clique to refer to the corresponding complete subgraph. Let
and
be two sets of vertices of
. We say that
is complete to
if every vertex in
is adjacent to every vertex in
, and that
is anticomplete to
if no vertex of
is adjacent to a vertex of
. A complete of three vertices is called a triangle.
The neighborhood of a vertex is the set
consisting of all the vertices which are adjacent to
. The closed neighborhood of
is
. A vertex of
is simplicial if
is a complete. Equivalently, a vertex is simplicial if it belongs to only one clique.
Given a graph , we will denote by
the set of cliques of
. Also, for every
, we will denote by
the set of cliques containing
. Finally, define
.
The chromatic number of a graph is the smallest number of colors that can be assigned to the vertices of
in such a way that no two adjacent vertices receive the same color, and is denoted by
. An obvious lower bound is the maximum cardinality of the cliques of
, the clique number of
, denoted by
.
A graph is perfect if
for every induced subgraph
of
. Perfect graphs were defined by Berge in 1960 [1] and are interesting from an algorithmic point of view: while determining the chromatic number and the clique number of a graph are NP-hard problems, they are solvable in polynomial time for perfect graphs [8].
A hole is a chordless cycle of length at least . An antihole is the complement of a hole. A hole or antihole is said to be odd if it consists of an odd number of vertices.
Given a graph , the clique graph
of
is the intersection graph of the cliques of
. A K-coloring of a graph
is an assignment of colors to the cliques of
such that no two cliques with non-empty intersection receive the same color (equivalently, a K-coloring is a coloring of
). A Helly K-complete of a graph
is a collection of cliques of
with common intersection. A Helly K-clique is a maximal Helly K-complete. The K-chromatic number and Helly K-clique number of
, denoted by
and
, are the sizes of a minimum K-coloring and a maximum Helly K-clique of
, respectively. It is easy to see by definition that
and that
. Also,
for any graph
. A graph
is coordinated if
for every induced subgraph
of
. Coordinated graphs were defined and studied in [3]. There are three main open problems concerning this class of graphs:
(i) find all minimal forbidden induced subgraphs for the class of coordinated graphs,
(ii) determine the computational complexity of finding the parameters and
for coordinated graphs and/or some of their subclasses, and
(iii) is there a polynomial time recognition algorithm for the class of coordinated graphs?
Recently, in [2] and [4], questions (i) and (ii) were answered partially. For question (iii), it is shown in [9] that the problem is NP-hard and it is NP-complete even when restricted to a subclass of graphs with . In this note, we answer a question related to these problems, which is: how many minimally non-coordinated graphs with
vertices and
are there? In particular, we show (algorithmic) operations for generating a family
of minimally non-coordinated graphs, of size
, such that every graph of the family has
vertices and
edges, for every
. It is not difficult to see that the operations we give are not enough for generating every minimally non-coordinated graph with
, so the question of how to generate every minimally non-coordinated graph is still open.
2. GENERATING NON-COORDINATED GRAPHS
It has been proved recently that perfect graphs can be characterized by two families of minimal forbidden induced subgraphs [7] and recognized in polynomial time [6].
Theorem 1 (Strong Perfect Graph Theorem [7]). Let be a graph. Then the following are equivalent:
(i) no induced subgraph of is an odd hole or an odd antihole.
(ii) is perfect.
Coordinated graphs are a subclass of perfect graphs [3]. Moreover, ,
,
,
are minimally non-coordinated [3, 5]. Therefore,
is not coordinated for all
.
In [2, 4] and manuscript [5], partial characterizations of coordinated graphs by minimal forbidden induced subgraphs were found. In these partial characterizations, the families of minimal forbidden induced subgraphs with vertices have
size for every
. Another partial characterization which is not difficult to prove (see [9] for a sharper result) is the following.
Theorem 2. Let be a graph such that
. Then the following are equivalent:
(i) is perfect.
(ii) does not contain odd holes.
(iii) is coordinated.
Corollary 3. Let be a graph with
. Then
is coordinated if and only if
does not contain odd holes and
.
The aim of this note is to show, for every , a family of minimally non-coordinated graph of size
such that every graph has
vertices and
edges. In order to define our families, we are going to use exchanger and preserver graphs which were defined in [9].
A graph exchanges colors between two different vertices
, or simply
is an exchanger between
, if
satisfies the following conditions:
(i) No induced subgraph of is an odd hole.
(ii) .
(iii) .
(iv) Every induced path between has odd length.
(v) In any 3-K-coloring of the cliques of
are colored with the three colors.
Vertices and
are called connectors. Call redundant to every simplicial vertex
such that
. We say that an exchanger
is a minimal exchanger if and only if
does not satisfy condition (v) for every non-redundant vertex
(although this is not the standard way to define minimality, this minimality is useful for "joining"
with other graphs, because redundant vertices of
may be needed so that the cliques of
are also cliques of the joined graph). Please note that conditions (i) and (iv) are hereditary and that if
satisfies (i) and (ii) but
does not satisfy (ii) then, by Theorem 2 ,
does not satisfy (v).
A graph preserves colors between a set of distinct vertices
, or simply
is a preserver between
, if
satisfies the following conditions:
(i) No induced subgraph of is an odd hole.
(ii) .
(iii) for every
.
(iv) Every induced path between has odd length for all
.
(v) In any 3-K-coloring of the cliques of
are colored with only one color.
Vertices are called connectors. We say that a preserver
is a minimal preserver if and only if
does not satisfy condition (v) for every non-redundant vertex
. Please note that conditions (i), (ii) and (iv) are hereditary and that condition (v) is satisfied only if condition (iii) is also satisfied.
Let be two graphs (
may be non-empty). The graph
has vertex set
and edge set
.
We say that graphs are compatible when
is a partition of
. We call them minimally compatible if they are compatible and
contains no redundant vertices.
Theorem 4. Let be an exchanger between
and
be a preserver between
such that
are compatible and
. Then
is non-coordinated. Moreover, if both
and
are minimal and minimally compatible then
is minimally non-coordinated.
Proof. Since is a partition of
and
by definition, then it follows that
,
for every
and
for every
. Consequently,
.
Suppose, contrary to our claim, that is coordinated. Then, there exists a
-K-coloring
of
. Since the cliques of
are also cliques of
, then the K-coloring
obtained by restricting the domain of
to the cliques of
is a
-K-coloring of
. Analogously, define
which is a
-K-coloring of
. Since
is an exchanger then there exist cliques
such that
for
where, w.l.o.g.,
and
. Also, since
is a preserver, there exist cliques
such that
where
and
. Therefore,
for
and
which is a contradiction because
and
are pairwise different,
and
. Consequently,
is a non-coordinated graph.
From now on, suppose that and
are both minimal and minimally compatible. Let us see that
contains no odd hole. On the contrary, suppose
contains an odd hole
. Since neither
nor
contains odd holes and
is anticomplete to
, then it follows that
can be partitioned into two paths
from
to
with disjoint interior and such that
is an induced path of
and
is an induced path of
. But this is impossible, because every induced path between
and
has odd length in both
and
, by definition.
Let be any proper induced subgraph of
; we have to prove that
. Since
contains no odd hole then, by Corollary 3, if
it follows that
is coordinated. Thus, it suffices to prove that if
then
which is equivalent to prove that
for every vertex
(because
for every induced subgraph
of
). We divide the proof into three cases:
Case 1: (
is analogous). If
then
. Otherwise, let
be a
-K-coloring of
where the cliques of
are colored using colors from the set
, and let
be a
-K-coloring of
where the clique to which
belongs has color
. Since
and
is connected it follows that the clique to which
belongs in
has at least one more vertex, thus it is still a clique in
. Therefore
is a valid
-K-coloring of
.
Case 2: . Suppose first that
is a redundant vertex of
, that is,
is simplicial and
. Since
(because
and
are minimally compatible) and
is simplicial in
, then it follows that
is a complete but not a clique of
. Since
is the disjoint union of
and
and
is a clique of
, it follows that
and
is complete in
. Hence
. Since
and
is connected, it follows that
is a triangle, consequently
and
. Now, suppose that
is not simplicial or
. Then,
has a
-K-coloring
where the clique containing
has color
and the clique containing
has color
(because
is minimal). Let
be a
-K-coloring of
where the cliques containing
have colors
and the cliques containing
have colors
. Then
is a valid
-K-coloring of
.
Case 3: . By minimality, if
is not redundant then
has a
-K-coloring
where the cliques of
are colored with at most two colors, say
. Let
be a
-K-coloring of
where the cliques containing
and
all have color
. Then
is a
-K-coloring of
. If
is redundant then, as before, it follows that
is the clique
in
and
belong to a clique
in
. Let
be a
-K-coloring of
where
has color
and
be a
-K-coloring of
where
has color
. Then
is a
-K-coloring of
.
In this section we define a recursive operator for constructing exponentially many non-isomorphic exchangers. Then we use these exchangers and one preserver to generate the exponential families, as in Section 2 . More operators for constructing exchangers and preservers are shown in the next section.
Let be vertices of a graph
and
be vertices of
. The graph
has
and
. In particular, when
is an exchanger between
,
is a preserver between
, and
, we denote
. Figure 2 shows examples of this operation, using graphs in Figure 1
Figure 1. Preservers and
and exchanger
. In every graph the connectors are
and
.
Figure 2. and
are shown. Both graphs are minimal exchangers with connectors
and
.
Lemma 5. Let be an exchanger between
and
be a preserver between
where
is not adjacent to
. Suppose also that
. Then
exchanges colors between
. Moreover, if both
and
are minimal then
is also minimal.
Proof. Arguments similar to those in Theorem 4 show that contains no odd hole and that
. Also, since
and
are cliques, it follows that
.
Let be an induced path between
. If
is also a path of
then it must have odd length. If
is not a path of
, then there must exists a subpath
which is a path of
between
such that
. Since
has odd length it follows that
also has odd length.
Suppose for a moment that is
-K-colorable and let
be a
-K-coloring of
. Let
,
and
,
be the cliques of
and
in
, respectively, and
and
be the cliques of
and
in
, respectively. Since
, and this union is disjoint, then it follows that the coloring
obtained by restricting the domain of
to the cliques of
is a
-K-coloring of
. Analogously, define
. Therefore, we may assume without loss of generality that
,
,
,
. Then
,
and by definition of
,
. Hence in every 3-K-coloring of
the cliques of
are colored with the three colors. By using the conditions derived for
, it is easy to see that
has at least one
-K-coloring as supposed, thus
.
From now on, suppose and
are both minimal. Let
. If
then
belongs to only one clique of
. Since
is a preserver, the clique of
has the same color as one of the cliques of
, therefore,
does not satisfy condition (v). The case
is analogous. If
is redundant in
, then it is also redundant in
. If
is a non-redundant vertex of
then, by minimality of
, there exists a K-coloring
which contradicts condition (v) for
. As in Theorem 4, it is easy to combine
with a K-coloring of
such that the coloring obtained is a K-coloring of
not satisfying condition (v). Similar arguments can be used to conclude the proof when
.
We are now ready to show the exponential families of minimally non-coordinated graphs. Let and
be the preserver graphs and
be the exchanger graph shown in Figure 1. Let
(
) denote the family of minimal exchangers defined by:
Both graphs of are shown in Figure 2. It is easy to see that graphs in
are pairwise non-isomorphic and that
. Also, by construction,
and
for every
. Finally, every exchanger in
is a minimal exchanger by Lemma 5 , and is minimally compatible with
(where connectors of
are identified with the connectors of the exchanger). Now, define
for every
. By Theorem 4 every graph in
is minimally non-coordinated, that is,
is one of the exponential families.
In this section we show other recursive operations for constructing exchangers and preservers. The motivation is to show how exchangers and preservers can be "joined" in a recursive manner. The proofs that these operations generate exchangers and preservers are left to the reader, and can be done in a similar way as the one in the previous section. Instead, we are going to draw sketches showing how vertices of the input graphs are tied together. The components of these sketches are shown in Figure 3 . To represent a preserver between
we are going to draw an oval labeled with P together with two points labeled
each one joined to the oval by a line. The oval represents the graph
, the points represent
and
and the line between
(
) and the oval represents the clique of
(
). In a similar manner, to represent an exchanger
between
we are going to draw an oval labeled with X together with two points labeled
each one joined to the oval by two lines. Again, the oval represents
, the points represent vertices
and the lines represent their cliques (in this case, one line of
and one of
may represent the same clique). Finally, a clique with
vertices is represented by an oval labeled with
and
points outside the oval, representing each of the
vertices. A line from one point to the oval means that the vertex belongs to the clique. Recall that a clique is a special kind of preserver, thus their vertices are also called connectors. Sometimes we also decorate the lines of the sketches with colors that represent a valid K-coloring.
Figure 3. Examples of sketches. On the left there is a component for an exchanger , in the middle there is a sketch component for a preserver
between
, and on the right there is a sketch for
.
Let be graphs which are preservers or exchangers between
and
, respectively, where
, and let
and
be sketches representing
and
, respectively. We are going to represent the graph
with a sketch formed by
and
, where for every vertex
of
, the corresponding points of
and
are drawn as a single point. One of such sketches is shown in Figure 3.
The sketches in Figure 4 represent preservers between two vertices , the sketches in Figure 5 represent exchangers between
and the sketch in Figure 6 represents a preserver between
. If the set of graphs of a sketch
are minimally compatible and minimal (as preservers or exchangers) then the graph represented by
is also minimal.
Figure 4. Two sketches of preservers between vertices and
.
Figure 5. Two sketches of exchangers between .
Figure 6. A sketch of a preserver between .
5. CONCLUSIONS AND FURTHER REMARKS
In this note we have shown one exponential-size family of minimally non-coordinated graphs for every natural number, and several operations for building preservers and exchangers. It is not difficult to define other operations for constructing minimal preservers or exchangers in order to generate different families of minimally non-coordinated graphs. Also, adding edges to some minimally non-coordinated graph may result into another minimally non-coordinated graph. Moreover, it is not clear that preservers and exchangers are enough to define every minimally non-coordinated graph. Perhaps a set of basic graphs together with operations for generating minimally non-coordinated graphs can be defined in a more convenient way.
With all these observations it seems difficult to find a characterization of coordinated graphs by minimal forbidden induced subgraphs, even when we restrict our attention to the class of graphs with . This is in turn a complementary result to that one in [9], which states that the problem of determining whether a graph in a very restricted subclass of graphs with
is coordinated is NP-complete.
[1] Claude Berge. Les problèmes de coloration en théorie des graphes. Publ. Inst. Statist. Univ. Paris, 9:123-160, 1960. [ Links ]
[2] Flavia Bonomo, Maria Chudnovsky, and Guillermo Durán. Partial characterizations of clique-perfect graphs. I. Subclasses of claw-free graphs. Discrete Appl. Math., 156(7):1058-1082, 2008. [ Links ]
[3] Flavia Bonomo, Guillermo Durán, and Marina Groshaus. Coordinated graphs and clique graphs of clique-Helly perfect graphs. Util. Math., 72:175-191, 2007. [ Links ]
[4] Flavia Bonomo, Guillermo Durán, Francisco Soulignac, and Gabriel Sueiro. Partial characterizations of clique-perfect and coordinated graphs: superclasses of triangle-free graphs. Discrete Appl. Math., 2009. In press. [ Links ]
[5] Flavia Bonomo, Guillermo Durán, Francisco Soulignac, and Gabriel Sueiro. Partial characterizations of coordinated graphs: line graphs and complements of forests. Math. Methods Oper. Res., 69(2):251-270, 2009. [ Links ]
[6] Maria Chudnovsky, Gérard Cornuéjols, Xinming Liu, Paul Seymour, and Kristina Vuskovic. Recognizing Berge graphs. Combinatorica, 25(2):143-186, 2005. [ Links ]
[7] Maria Chudnovsky, Neil Robertson, Paul Seymour, and Robin Thomas. The strong perfect graph theorem. Ann. Math. (2), 164(1):51-229, 2006. [ Links ]
[8] M. Grötschel, L. Lovász, and A. Schrijver. The ellipsoid method and its consequences in combinatorial optimization. Combinatorica, 1(2):169-197, 1981. [ Links ]
[9] Francisco Soulignac and Gabriel Sueiro. Np-hardness of the recognition of coordinated graphs. Ann. Oper. Res., 169(1):17-34, 2009. [ Links ]
Francisco Soulignac
Departamento de Computación
Facultad de Ciencias Exactas y Naturales
Universidad de Buenos Aires
Ciudad de Buenos Aires, Argentina
fsoulign@dc.uba.ar
Gabriel Sueiro
Departamento de Computación
Facultad de Ciencias Exactas y Naturales
Universidad de Buenos Aires
Ciudad de Buenos Aires, Argentina
gsueiro@dc.uba.ar
Recibido: 6 de agosto de 2007
Aceptado: 10 de julio de 2008