1 Introduction

Almost every graph is divergent under the biclique operator

Marina Groshaus 111Partially supported by UBACyT grant 20020100100754, PICT ANPCyT grant 2010-1970, CONICET PIP grant 11220100100310222Partially supported by Math-Amsud project 14 Math 06

Universidad de Buenos Aires

Departamento de Computación

groshaus@dc.uba.ar


André L.P. Guedes22footnotemark: 2

Universidad de Buenos Aires / Universidade Federal do Paraná

Departamento de Computación / Departamento de Informática

andre@inf.ufpr.br


Leandro Montero

Universidad de Buenos Aires / Université Paris-Sud

Departamento de Computación / Laboratoire de Recherche en Informatique

lmontero@{dc.uba.ar/lri.fr}




ABSTRACT

A biclique of a graph is a maximal induced complete bipartite subgraph of . The biclique graph of denoted by , is the intersection graph of all the bicliques of . The biclique graph can be thought as an operator between the class of all graphs. The iterated biclique graph of denoted by , is the graph obtained by applying the biclique operator successive times to . The associated problem is deciding whether an input graph converges, diverges or is periodic under the biclique operator when grows to infinity. All possible behaviors were characterized recently and an algorithm for deciding the behavior of any graph under the biclique operator was also given. In this work we prove new structural results of biclique graphs. In particular, we prove that every false-twin-free graph with at least vertices is divergent. These results lead to a linear time algorithm to solve the same problem.

Keywords: Bicliques; Biclique graphs; False-twin-free graphs; Iterated graph operators; Graph dynamics

1 Introduction

Intersection graphs of certain special subgraphs of a general graph have been studied extensively. For example, line graphs (intersection graphs of the edges of a graph), interval graphs (intersection of intervals of the real line), clique graphs (intersection of cliques of a graph), etc [4, 5, 10, 13, 14, 28, 30].

The clique graph of denoted by , is the intersection graph of the family of all maximal cliques of . Clique graphs were introduced by Hamelink in [20] and characterized by Roberts and Spencer in [35]. The computational complexity of the recognition problem of clique graphs had been open for more that 40 years. In [1] they proved that clique graph recognition problem is NP-complete.

The clique graph can be thought as an operator between the class of all graphs. The iterated clique graph is the graph obtained by applying the clique operator successive times (). Then is called clique operator and it was introduced by Hedetniemi and Slater in [21]. Much work has been done on the scope of the clique operator looking at the different possible behaviors. The associated problem is deciding whether an input graph converges, diverges or is periodic under the clique operator when grows to infinity. In general it is not clear that the problem is decidable. However, partial characterizations have been given for convergent, divergent and periodic graphs restricted to some classes of graphs. Some of these lead to polynomial time recognition algorithms. For the clique-Helly graph class, graphs which converge to the trivial graph have been characterized in [3]. Cographs, -tidy graphs, and circular-arc graphs are examples of classes where the different behaviors are characterized [7, 23]. Divergent graphs were also considered. For example in [32], families of divergent graphs are shown. Periodic graphs were studied in [10, 27]. In particular it is proved that for every integer , there exist periodic graphs with period and also convergent graphs which converge in steps. More results about iterated clique graph can be found in [11, 12, 24, 25, 26, 33].

A biclique is a maximal bipartite complete induced subgraph. Bicliques have applications in various fields, for example biology: protein-protein interaction networks [6], social networks: web community discovery [22], genetics [2], medicine [31], information theory [19], etc. More applications (including some of these) can be found in [29].

The biclique graph of a graph denoted by , is the intersection graph of the family of all maximal bicliques of . It was defined and characterized in [16]. However no polynomial time algorithm is known for recognizing biclique graphs. As for clique graphs, the biclique graph construction can be viewed as an operator between the class of graphs.

The iterated biclique graph is the graph obtained by applying to the biclique operator times iteratively. It was introduced in [15] and all possible behaviors were characterized. It was proven that a graph is either divergent or convergent but it is never periodic (with period bigger than ). In addition, general characterizations for convergent and divergent graphs were given. These results were based on the fact that if a graph contains a clique of size at least , then or contains a clique of larger size. Therefore, in that case diverges. Similarly if contains the or the graphs as an induced subgraph, then contains a clique of size , and again diverges. Otherwise it was shown that after removing false-twin vertices of , the resulting graph is a clique on at most vertices, in which case converges. Moreover, it was proved that if a graph converges, it converges to the graphs or , and it does so in at most steps. These characterizations leaded to an time algorithm for recognizing convergent or divergent graphs under the biclique operator.

In this work we show new results that lead to a linear time algorithm to solve the same problem. We study conditions for a graph to contain a , a , a , a or a (see Figure 1) as induced subgraphs so we can decide divergence (since ). First we prove that if has at least bicliques then it diverges. Then, we show that every false-twin-free graph with at least vertices has at least bicliques, and therefore diverges. Since adding false-twins to a graph does not change its behavior, then the linear algorithm is based on the deletion of false-twin vertices of the graph and looking at the size of the remaining graph.

Figure 1: Graphs , , , and , respectively.

It is worth to mention that these results are indeed very different from the ones known for the clique operator, for which it is still an open problem to know the computational complexity of deciding the behavior of a graph under the clique operator.

This work is the full version of a previous extended abstract published in [17]. It is organized as follows. In Section the notation is given. Section contains some preliminary results that we will use later. In Section we prove that any graph with at least bicliques diverges, and that every graph with at least vertices with no false-twins vertices contains at least bicliques. This leads to a linear time algorithm to decide convergence or divergence under the biclique operator.

2 Notation and terminology

Along the paper we restrict to undirected simple graphs. Let be a graph with vertex set and edge set , and let and . A subgraph of is a graph where and . A subgraph of is induced when for every pair of vertices , if and only if . A graph is -free if it does not contain as an induced subgraph. A graph is bipartite when , and . Say that is a complete graph when every possible edge belongs to . A complete graph of vertices is denoted . A clique of is a maximal complete induced subgraph while a biclique is a maximal bipartite complete induced subgraph of . The open neighborhood of a vertex denoted , is the set of vertices adjacent to while the closed neighborhood of denoted by , is . Two vertices , are false-twins if . A vertex is universal if it is adjacent to all of the other vertices in . A path (cycle) of vertices, denoted by (), is a set of vertices such that for all and is adjacent to for all (and is adjacent to ). A graph is connected if there exists a path between each pair of vertices. We assume that all the graphs of this paper are connected.

A is a complete graph with vertices and a vertex adjacent to two of them. A is the graph obtained by joining two copies of the with a common vertex.

Given a family of sets , the intersection graph of is a graph that has the members of as vertices and there is an edge between two sets when and have non-empty intersection.

A graph is an intersection graph if there exists a family of sets such that is the intersection graph of . We remark that any graph is an intersection graph [37].

A family of sets is mutually intersecting if every pair of sets have non-empty intersection.

Let be any graph operator. Given a graph , the iterated graph under the operator is defined iteratively as follows: and for , . We say that a graph diverges under the operator whenever . We say that a graph converges under the operator whenever for some , that is, for every and some . We say that a graph is periodic under the operator whenever for some , .

The iterated biclique graph is the graph obtained by applying iteratively the biclique operator times to .

In the paper we will use the terms convergent or divergent meaning convergent or divergent under the biclique operator .

By convention we arbitrarily say that the trivial graph is convergent under the biclique operator (observe that this remark is needed since the graph does not contain bicliques).

3 Preliminary results

We start with this easy observation.

Observation 3.1 ([15]).

If is an induced subgraph of , then is a subgraph (not necessarily induced) of .

The following proposition is central in the characterization of convergent and divergent graphs under the biclique operator. Basically, it shows that if a graph contains a big complete subgraph, it is going to grow in one or two steps of .

Proposition 3.2 ([15]).

Let be a graph that contains as a subgraph, for some . Then, or .

Next theorem characterizes the behavior of a graph under the biclique operator.

Theorem 3.3 ([15]).

If contains either or the or the as an induced subgraph, then is divergent. Otherwise, converges to or in at most 3 steps.

Notice that differently than the clique operator, a graph is never periodic under the biclique operator (with period bigger than 1). We remark the importance of the graph to decide the behavior of a graph under the biclique operator since we have that and .

Observe that as proved in [15], the biclique graph does not change by the deletion or addition of false-twin vertices since each pair of false-twins belongs to exactly the same set of bicliques. That is, for any graph , for any false-twin vertex . It follows that the behavior of a graph under does not change either. Therefore we focus our study on false-twins-free graphs. For that we need the following definition used in [15].

Consider all maximal sets of false-twin vertices and let be the set of representative vertices such that . The graph obtained by the deletion of all vertices of for , is denoted . Observe that has no false-twin vertices.

Using , as a corollary of Theorem 3.3, the next useful result was obtained.

Corollary 3.4 ([15]).

A graph is convergent if and only if has at most four vertices. Moreover, for .

We recall that the number of bicliques of a graph may be exponential in the number of its vertices [34]. However, if some vertex of a graph lies in five bicliques, then contains a thus diverges. If every vertex of belong to at most four bicliques, then has at most bicliques. Therefore, since each biclique can be generated in  [8, 9], constructing takes . Building can be done in time using the modular decomposition [18]. From Corollary 3.4, if has at most four vertices, then is convergent, otherwise is divergent. Hence the overall algorithm runs in time.

4 Linear time algorithm

In this section we give a linear time algorithm for deciding whether a given graph is divergent or convergent under the biclique operator.

Motivated by Theorem 3.3 and Corollary 3.4, we study structural properties of a graph to guarantee that its biclique graph contains and therefore diverges.

The following two lemmas answer that question.

Lemma 4.1.

Let for some graph . Let , be false-twin vertices of and , their associated bicliques in . Suppose that there are no edges between vertices of and vertices of . Then there exists a vertex such that is adjacent to every vertex of and . Furthermore, contains a as induced subgraph.

Proof.

Let , be false-twin vertices of and , their associated bicliques in , such that there are no edges between vertices of and vertices of . Since is connected, take the shortest path from some vertex of to . Let be the first vertex in the path such that . Clearly, . Let be a vertex adjacent to .

First, suppose that there exists a vertex such that is not adjacent to . Consider the following alternatives:

Case 1: . Then is contained in some biclique , and , such that it does not intersect since there is no edge between and . This is a contradiction since and are false-twin vertices. It follows that every vertex in not adjacent to is not adjacent to .

Case 2: . Then there exists a vertex adjacent to and . By case 1, must be adjacent to . This is the same situation as previous case but considering instead of and the biclique containing instead of . A contradiction.

We conclude that for all , is adjacent to .

Now, the edge is contained in a biclique that must intersect as are false-twin vertices of . Since there are no edges between and there exists a vertex such that is adjacent to . The same argument used for and also holds for and . That is, for all , is adjacent to .

Figure 2: Bicliques and with new bicliques containing .

Finally, let be adjacent vertices in and let be adjacent vertices in . Since are adjacent to , then , , and are contained in four different bicliques , , and such that , for . As , for (Fig. 2), is an induced subgraph of . ∎

Lemma 4.2.

Let for some graph . Let be false-twin vertices of and let be their associated bicliques in . Suppose that for any pair of bicliques , , there is an edge between some vertex of and some vertex of . Then, is an induced subgraph of .

Proof.

Let be the false-twin vertices of and their associated bicliques in such that for any pair of bicliques , , there is an edge between some vertex of and some vertex of . We will show that contains either a , a , a or a , or four mutually intersecting bicliques also intersecting with , and . In any case we obtain a in . We have the following cases:

Case 1: There is a with one vertex in each biclique. Let , , be the . Now , and are contained in different bicliques of . It is easy to see that none of , or are bicliques isomorphic to , otherwise they would not intersect the biclique containing the opposite edge of the (e.g. with ) contradicting that are false-twin vertices.

Case 1.1: One of the bicliques, say , is isomorphic to where the vertex is in the partition of size one. As the biclique containing must intersect , there exists a vertex adjacent to and not adjacent to . Now, as , there exists a vertex , such that is adjacent to . Therefore induces a or a depending on the edge . See Figure 3.

Figure 3: Case 1.1

Case 1.2: None of the bicliques , and are isomorphic to where the vertex of the is in the partition of size one. As the biclique containing has to intersect , call a vertex in that intersection and w.l.g. assume adjacent to and not to .

Case 1.2.1: Suppose is adjacent to . Now, as is not isomorphic to , we have the following cases.

If there exists a vertex adjacent to and not adjacent to . Depending on the edge , induces a or , , and are contained in four mutually intersecting bicliques. See Figure 4.

Figure 4: Case 1.2.1 with adjacent to and not to

Otherwise, assuming that every adjacent to is adjacent to , and considering that , there exists adjacent to and . In this case induces a or a depending on the edge . See Figure 5.

Figure 5: Case 1.2.1 with adjacent to and

Case 1.2.2: There exists not adjacent to and , and adjacent to . Let be any vertex adjacent to (and consecuently to ). Clearly, if is adjacent to , it must be adjacent to , otherwise we would be in the case above. So, if is adjacent to both, induces a . Therefore, we can assume that for every adjacent to and , is not adjacent to and . Moreover, this must be also true for every vertex in adjacent to and every vertex in adjacent to , that is, every vertex in adjacent to is not adjacent to and , and every vertex in adjacent to is not adjacent to and . Suppose that there exists adjacent to and not adjacent to , then , , and are contained in four mutually intersecting bicliques. Then, we can assume is adjacent to . Indeed, assume that every vertex in adjacent to is adjacent to every vertex in adjacent to and to every vertex in adjacent to . Also every vertex in adjacent to is adjacent to every vertex in adjacent to . Otherwise, we would obtain four mutually intersecting bicliques. Let adjacent to . Observe that if is adjacent to then is also adjacent to , otherwise we are in case 1.2.1 considering the . Then, depending on the edge , induces a , or , , and are contained in four mutually intersecting bicliques. See Figure 6.

Figure 6: Case 1.2.2

We covered all the cases when a is in .

Case 2: There is an induced in such that , and , that is, . Now as , there exists either adjacent to and , or adjacent to and not adjacent to . We have the following cases:

Case 2.1: is adjacent to and (the case where is adjacent to and is analogous). Observe that is not adjacent to as we would obtain a triangle with one vertex in each biclique (case 1). Let be a vertex adjacent to . If is adjacent to then induces a (otherwise case 1, considering and , or and , adjacent vertices). Then assume every vertex adjacent to is not adjacent to . Furthermore, if any vertex adjacent to , is also adjacent to , then induces a , a or a depending on the edges , . Therefore we can assume that every vertex adjacent to is not adjacent to . See Figure 7.

Figure 7: Case 2.1

Case 2.1.1: There is some not adjacent to . Now as , there exists adjacent to and not adjacent to . If is adjacent to then induces a . We can assume is not adjacent to .

If is adjacent to then , , and one of or depending on the edge , are contained in four different mutually intersecting bicliques. So we can assume is not adjacent to .

As , either is not adjacent to some vertex of that is adjacent to , or forms a triangle with two vertices of .

Suppose first that is not adjacent to such that is adjacent do . Note that , and are contained in three different mutually intersecting bicliques. See Figure 8.

Figure 8: Case 2.1.1 with not adjacent to

If is not adjacent to then is contained in the fourth biclique (and we got four different mutually intersecting bicliques). So suppose is adjacent to . If is not adjacent to , the fourth biclique contains . Finally, if is adjacent to then is contained in the fourth biclique.

Suppose next that forms a triangle with two vertices of . That is, there are two adjacent vertices such that is adjacent to and , and is adjacent to (see Figure 9). If is adjacent to , then depending on the edge , induces a or a . Assume therefore that is not adjacent to . Then, , and depending on the edge , either and , or and are contained in four different mutually intersecting bicliques.

Figure 9: Case 2.1.1 form a triangle with 2 vertices of

Case 2.1.2: Every vertex adjacent to is adjacent to . Now as , there exists adjacent to and . Note that is not adjacent to , otherwise case 1. Then induces a , or depending on the edges and . See Figure 10.

Figure 10: Case 2.1.2

Case 2.2: is adjacent to and not adjacent to . By symmetry there exists adjacent to and not adjacent to . Assume that is not adjacent to and is not adjacent to (otherwise case 2.1).

Suppose first that there exists adjacent to and . Observe that is not adjacent to (case 1 considering the ) and is not adjacent to and to at the same time (case 2.1 considering the ). Depending on the edge , one of or along with , , are contained in four different mutually intersecting bicliques.

Figure 11: Case 2.2, with edge and without edge

Suppose therefore that every adjacent to is not adjacent to . If is not adjacent to or is adjacent to , then , , and are contained in four different mutually intersecting bicliques. See Figure 11. Otherwise, is adjacent to and not adjacent to . Consider the , where the edge is contained in , vertex and . Now, following the same arguments as above, considering vertex as , vertex as , and vertex as , since the vertex (that is adjacent to and not adjacent to and ) has the same “role” as the vertex , we arrive exactly to the previous case (when is not adjacent to or is adjacent to , Figure 11). Therefore , , and are contained in four different mutually intersecting bicliques, where is adjacent to and not adjacent to nor to , and is adjacent to and not adjacent to nor to . See Figure 12.

Figure 12: Case 2.2, with

We covered all the cases when a is in with all of the vertices in the bicliques , and .

Case 3: There is an induced , in with at least one vertex from each biclique , and . For the case there is nothing to do. Finally, for , it is easy to see that, as each biclique containing two consecutive edges of the has to intersect , and , then we would obtain a smaller cycle and therefore this case cannot occur.

Since we covered all cases the proof is done.

Next, we present the main theorem of this section. This theorem shows that almost every graph is divergent under the biclique operator. We remark that the linear time algorithm for recognizing convergent or divergent graphs given later in this section is based on this theorem.

Theorem 4.3.

Let be a graph. If has at least bicliques, then diverges under the biclique operator.

Proof.

By way of contradiction, suppose that has at least bicliques and converges under the biclique operator. By Corollary 3.4, for . Consider the following cases.

Case . Then is a contradiction since has at least bicliques.

Case . In [16] it was proved that no bipartite graph with more than two vertices is a biclique graph. Then what means that has only bicliques and therefore a contradiction.

Case . Since has at least bicliques it follows that in there exists a set of false-twin vertices of size at least three. Consider the bicliques of associated to the three false-twin vertices. If there is a pair of bicliques such that there is no edge between any vertex of and any vertex of , by Lemma 4.1 it follows that is an induced subgraph of . Otherwise, for every two pair of bicliques there is an edge between some vertex of and some vertex of and by Lemma 4.2 contains as an induced subgraph. In any case, by Theorem 3.3 diverges under the biclique operator, a contradiction.

Case . There are two alternatives. Suppose that has a set of false-twin vertices of size at least three. Then following the proof of the case we arrive to a contradiction. Otherwise, there are only two possible graphs isomorphic to ( has or vertices and it has no set of three false-twin vertices, see Fig. 13). By inspection, using the characterization of biclique graphs given in [16], we prove that these two graphs are not biclique graphs. We conclude that this case cannot occur.

Figure 13: Unique two possible graphs for case .

Since we covered all cases, diverges under the biclique operator and the proof is finished. ∎

The next step is to study graphs without false-twin vertices with at least bicliques. This will complete the idea of the linear time algorithm for recognizing divergent and convergent graphs under the biclique operator.

Theorem 4.4.

Let be a false-twin-free graph. If has at least vertices then has at least bicliques.

Proof.

We prove the result by induction on . For , by inspection of all graphs without false-twin vertices the result holds. Suppose now that . Theorem in [36] states that if a graph has no false-twin vertices, then there exists a vertex such that is also false-twin free. Consider such a vertex and let . If is connected, since it has at least vertices, by the inductive hypothesis it has at least bicliques. Now as is an induced subgraph of we conclude that also has at least bicliques. Suppose now that is not connected. Let be the connected components of on vertices respectively. Since has no false-twin vertices, it can be at most one such that . If there is one component with at least vertices, then by the inductive hypothesis this component has at least bicliques and so does . Therefore every component has at most vertices. Now, by inspection we can verify that every component (but maybe one with just vertex) has at least bicliques. Also, since is disconnected, along with at least one vertex of each of the components is a biclique in isomorphic to that is lost in . Summing up and assuming the worst case, that is, there exists one (suppose ) we obtain that the number of bicliques of is at least

as we wanted to prove. Now the proof is complete. ∎

Theorem 4.4 implies that the number of convergent graphs without false-twin vertices is finite since convergent graphs without false-twin vertices have at most vertices. This fact leads to the following linear time algorithm.

Algorithm: Given a graph , build . If has at least vertices, answer “ diverges” and STOP. Otherwise, build . If has at most vertices answer “ converges” and STOP. Otherwise, answer “ diverges” and STOP.

The algorithm has time complexity. For this observe that can be built in time using the modular decomposition [18]. Finally, if has at most vertices any further operation takes time complexity.

5 Conclusions

In [15] it is given an time algorithm to recognize convergent and divergent graphs under the biclique operator. In this paper we prove that graphs without false-twin vertices with at least vertices diverge. This shows that “almost every” graph is divergent and as a direct consequence, we obtain a linear time algorithm for recognizing the behavior of a graph under the biclique operator. We remark that in contrast as the iterated clique operator, no polynomial time algorithm is known for recognizing any of its possible behaviors.

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