versão impressa ISSN 0041-6932
Rev. Unión Mat. Argent. vol.50 no.1 Bahía Blanca jun. 2009
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.
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  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 .
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 . 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  and , questions (i) and (ii) were answered partially. For question (iii), it is shown in  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.
Theorem 1 (Strong Perfect Graph Theorem ). 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.
In [2, 4] and manuscript , 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  for a sharper result) is the following.
(i) is perfect.
(ii) does not contain odd holes.
(iii) is coordinated.
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 .
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.
(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.
(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 .
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 .
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 , which states that the problem of determining whether a graph in a very restricted subclass of graphs with is coordinated is NP-complete.
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Departamento de Computación
Facultad de Ciencias Exactas y Naturales
Universidad de Buenos Aires
Ciudad de Buenos Aires, Argentina
Departamento de Computación
Facultad de Ciencias Exactas y Naturales
Universidad de Buenos Aires
Ciudad de Buenos Aires, Argentina
Recibido: 6 de agosto de 2007
Aceptado: 10 de julio de 2008