A biclique of a graph is a vertex set that induces a maximal complete bipartite subgraph. The biclique graph of a graph , denoted by , is the intersection graph of the bicliques of . The biclique graph was introduced by Groshaus and Szwarcfiter , based on the concept of clique graphs. They gave a characterization of biclique graphs (in general) and a characterization of biclique graphs of bipartite graphs. The time complexity of the problem of recognizing biclique graphs remains open.
Bicliques in graphs have applications in various fields, for example, biology: protein-protein interaction networks , social networks: web community discovery , genetics , medicine , information theory . More applications (including some of these) can be found in the work of Liu, Sim and Li .
The biclique graph can be considered as a graph operator: given a graph , the operator returns the biclique graph of , [9, 8, 13]. Some problems related to graph operators are studied in relation to some classes of graphs. Given a graph operator and a class , it is studied the problem of recognizing the class , or the problem of recognizing the class . These problems have been studied in the context of the clique graph, (clique operator). There are works about , where is clique-Helly, chordal, interval, split, diamond-free, dismantable graphs, arc-circular graphs etc [2, 6, 16, 17, 21].
In this work we prove that every biclique graph of a -free graph (triangle-free) is the square of some graph, that is, -free, where denote the class of all graphs and denote the class of the square graphs of the graphs of the class . This result gives a tool for studying other classes of biclique graphs. Known results on square graphs follow directly for biclique graphs (using known properties of square graphs). Also, some results on biclique graphs of graphs, when restricted to -free graphs, can be easily proven using the fact that it is a square graph. For example, the fact that every is contained in a diamond or a -fan , the fact that the number of vertices of degree is at most and the fact that the family of neighbourhoods of vertices of degree 2 satisfy the Helly property .
Groshaus and Montero presented a conjecture stating that a certain structure is forbidden in biclique graphs [10, Conjecture 6.2]. We prove that this conjecture holds when is a -free graph, using the fact that is the square of some particular graph.
Conjecture 1 ().
If for some graph , where is not isomorphic to the diamond then there do not exist such that and their neighbours are contained in a for .
A comparability graph is such that there is a partially ordered set (poset) where if and only if and are comparable by . We define a subclass of comparability graphs, the class of interval intersection closed comparability (IIC-comparability) graphs, and we prove that the class of biclique graphs of bipartite graphs is equal to the class of the square graphs of IIC-comparability graphs, that is, bipartiteIIC-comparability, giving another characterization of biclique graphs of bipartite graphs.
1.1 Some Definitions and Notations
Let be a graph with vertex set and edge set . Denote the set of neighbours of a vertex as . Let and define to be the set of vertices of that are incident to every vertex of . That is, Note that if , .
Given a poset and , let and be, respectively, the predecessors interval and successors interval of in . We say that a poset is interval intersection closed (IIC) if the sets and are closed under intersection. That is, for every pair , , the following sentences are true:
if then there is a such that );
if then there is a such that ).
Let the graph class IIC-comparability (Interval Intersection Closed Comparability) be the class of comparability graphs with posets that are IIC.
2 Mutually Included Biclique Graph
Denote a biclique of , with bipartition , as . That is, . Given a biclique and a vertex then (i) or (ii) there are vertices and such that . Note that in case (ii) the vertices , and form a . So, if is a -free graph (ii) is always false and if then (i) holds.
Given a -free graph , two independent sets, and , of form a biclique of iff and .
Note that no part of a biclique intersects both parts of another biclique. That is, if and are two bicliques of a graph , or . So, assume that and . We say that and are mutually included if and . See Figure 0(a).
Define , the mutually included biclique graph of , as the graph with the bicliques of as its vertex set and is an edge iff and are mutually included. Note that (with the same vertex set).
If and are two bicliques of a -free graph , that are not mutually included, such that then there is a biclique such that and .
Let be a -free graph and let and be two bicliques of that are not mutually included and that . Let , . Note that is an independent set (as is a -free graph) and that (by definition of ). See Figure 0(b).
Suppose is not a biclique of . Then, by Observation 1, . By definition, , so there is a vertex . As is neighbour of every vertex of , it is also neighbour of every vertex of , then induces a complete bipartite subgraph and is not a biclique. So, there is no such vertex , and is a biclique of . ∎∎
If and are two intersecting bicliques of a -free graph then and are mutually included or there is a biclique that is mutually included both with and .
If and are two bicliques such that there is a biclique mutually included with both and , then .
Suppose and . Let be a biclique mutually include with and . Suppose w.l.o.g. that and . Also suppose that . If , then and . On the other hand, if , then and and also . ∎∎
If is a -free graph, then .
Let be a -free graph and let and be two bicliques.
If and intersect, by Corollary 1, and are mutually included or there is a biclique that is mutually included with both. That is, and are at distance at most in .
If and do not intersect, by Lemma 2 there is no biclique that is mutually included with both. That is, and are at distance at least in .
So . ∎∎
By Theorem 1, -free.
In this section we present properties of mutually included biclique graphs, square graphs, and therefore, for biclique graphs of triangle-free graphs.
Let and be two bicliques of a -free graph such that . and are mutually included iff or .
Let and be two bicliques such that .
By definition if and are mutually included then or .
Now suppose . Then , by the definition. By Observation 1, , and . So, . Consequently and are mutually included.
Changing roles of and , we conclude that and are also mutually included in the case when . ∎∎
Note that, for any graph, the union of two mutually included bicliques induces a bipartite graph and a set of mutually included bicliques are “nested”. In Figure 3 are presented a set of “nested” mutually included bicliques.
For any graph , and a set of bicliques of such that every two of them are mutually included, then there is an ordering of , such that (and ), assuming that , for .
Let be any graph and be a set of bicliques of such that every two of them are mutually included. Recall that the relation is a partial order and note that the bicliques of has parts , , that are all comparable under that partial order. So there is an ordering of , such that (and ), assuming that , for . ∎∎
Any other biclique that is mutually included with , for some is also mutually included with , for every or every .
For every graph and for every clique of there is an ordering of the vertices of such that for every vertex , with , the vertices of are consecutive in that ordering and include the first or the last vertex of that ordering.
Let be a graph and a clique of . That is, is a set of bicliques of such that every two of them are mutually included. By Lemma 4, there is an ordering of the bicliques of . Let , such that . Suppose that and , for .
Suppose that for some and that is minimum. Then for every . That is, is mutually included with , for .
Now suppose that for some and that is maximum. By the same argument, is mutually included with , for . ∎∎
Using Theorem 2 can be proved that some structures do not occur in .
Let be the vertices of the triangle of the net graph (Figure 1(a)) and let be the other vertices such that each is an edge. Let be any graph generated by a net graph with 0 or more edges added connecting only vertices of . See Figure 4.
For every graph , does not contain a as an induced subgraph.
Let be a graph and . Suppose there is a graph as an induced subgraph of . As each vertex of is adjacent to only one vertex of , and is a triangle, by Theorem 2 there is an ordering of . Considering the neighbourhoods of and , and are the first and the last (or the inverse) of that ordering. Also by Theorem 2, the neighbourhood of should include or , but . Therefore does not contain a as an induced subgraph. ∎∎
Now we present a property about square graphs that implies in a property of biclique graphs of triangle-free graphs.
Call “old” edges the edges of (for some graph ) that are also edges of and call the other edges of as “new” edges. That is, “new” edges are the edges that are in and do not exist in .
Let for some graph . Then, every induced of is contained in a or a .
Let for some graph . Suppose induces a in . Observe first that can contain at most one old edge. If is a new edge, then there is a vertex of that is adjacent in to and to . Observe that can not be adjacent to any , in . Then, for every new edge of there is in a (different) vertex adjacent to and .
If all edges of are new edges, vertices , , induce a in . Consequently is contained in .
Otherwise, w.l.o.g. suppose is an old edge of . Then, , , is isomorphic to and is included in . ∎∎
That property (of biclique graphs of triangle-free graphs) is a generalization of a very used property involving in biclique graphs. Also, we think that this new property can be a useful tool for solving the biclique graph recognition problem.
Every induced of a biclique graph of triangle-free graph is contained in a or a .
Note that a is a , is a -fan and is a diamond. Then, the fact that every is contained in a diamond or a -fan , when restricted to biclique graphs of triangle-free graphs, is a particular case of Corollary 4.
We present the following result which generalizes the Conjecture 1 when restrict to biclique graph of triangle-free graphs.
Let be a biclique graph of a triangle-free graph. If there exists an independent set in with , such that . Then .
By Theorem 1, for some triangle-free graph . Suppose there is an independent set in with , such that .
Consider vertex . We affirm that there exists an old edge , . Let be a neighbour of . If is not an old edge, there exists such that , are edges in , that is and are old edges. Observe that when , since two vertices of does not have a common neighbour in , otherwise they would be adjacent in . Therefore .
Suppose there are only one old edge incident to each vertex of , otherwise . Suppose w.l.o.g. that . That is, , and are old edges. Since exists and is a new edge then, there exists a vertex such that and are old edges. Note that as is the unique old edge incident to and we conclude that is an old edge. Now following the same arguments for edges and we obtain that , and are old edges. Finally, vertices induce a in what leads to a contradiction according to Corollary 3. Consequently, . ∎∎
Observe that in Theorem 4 we do not ask for vertices to have the same neighbourhood. Finally, we prove the Conjecture 1 for the case of biclique graphs of triangle-free graphs as a corollary of Theorem 4.
Let be a graph not isomorphic to the diamond. Suppose , are vertices of with , such that , with , for . Then is not a biclique graph of a triangle-free graph .
4 Mutually Included Biclique Graphs of Bipartite Graphs
In the case of bipartite graphs the parts of the bicliques are also bipartitioned, and we can establish a partial order of the bicliques based on one part of the bipartite graph.
Let be a bipartite graph. Let be the set of bicliques of . Define the relation over such that when . The reflexive closure of is the partial order .
For every bipartite graph the poset is IIC.
Let be a bipartite graph, with the poset and let and be two different bicliques of .
If then and is the maximum of . Also and is the minimum of .
Now suppose and are not comparable.
If then .
Suppose and there is a biclique . By definition, and , that is, . Then, and . So, if then .
If then, by Lemma 1, there is a biclique such that and is the maximum of . The same for part and .
So, the poset is IIC. ∎∎
We show that bipartite is an IIC-comparability graph.
If is a bipartite graph then, is an IIC-comparability graph.
Let be a bipartite graph with the poset . By Lemma 3, two different bicliques, and , of are mutually included iff and are comparable by .
So, is the comparability graph of the poset .
Moreover, as is IIC, by Lemma 5, is an IIC-comparability graph. ∎∎
Let be a poset. Define the predecessors-successors bipartite graph as follows: ; ; , for .
Let and .
Given a poset and its predecessors-successors graph . For every , is a biclique of .
Let be a poset and its predecessors-successors graph .
Let . As for every , and for every , , then and . So induces a complete bipartite subgraph of .
Now suppose is not maximal. Then w.l.o.g. there is a vertex such that induces a complete bipartite subgraph of . By definition, , for every . But , and then . Consequently, and is a biclique of . ∎∎
For every subset , define or to be the base set of .
Given an IIC poset and its predecessors-successors graph . Every biclique of is equal to for some .
Let be an IIC poset and its predecessors-successors graph . And let be any biclique of .
Consider the sets and . Suppose do not have a maximum element. Then there are at least two maximal elements in , and . As is IIC and (as and have at least one neighbour in common in ), has a minimum, . As , , consequently . Moreover, and , which implies that and are not maximal elements of . So, has a maximum. Using a similar reasoning we can show that has a minimum. Let be the maximum element of . Then and is the minimum element of . That is, . ∎∎
We show now that the class of IIC-comparability is exactly the class of bipartite.
By Lemma 6, bipartite IIC-comparability.
Now let be an IIC-comparability graph, , with its IIC poset, , , and its predecessors-successors graph . Recall that is a bipartite graph.
Let , with .
Suppose . Then or (as is a comparability graph with poset ). So or and and are mutually included (by Lemma 3) and then .
Now suppose . Then or . As and , then or and .
So, is an isomorphism and . Consequently IIC-comparability bipartite.
Therefore, bipartite IIC-comparability. ∎∎
5 Final Remarks
In this work we prove that the biclique graph of a -free graph is the square of some particular graph called extending all known properties of square graphs to biclique graphs of triangle-free graphs and providing a tool to prove other properties. Some published properties about biclique graphs can be easily proved if we restrict to square graphs.
We prove that and by presenting the graphs -wheel (Figure 1(c)), that is the biclique graph of some graph but is not the square of any graph, and the square graph of the net (Figure 1(a)), that is not a biclique graph of any graph. We conclude that the biclique graphs of -free graphs are in the intersection , but it is not known if -free or -free.
Moreover, we give the first known property (which was conjectured by Groshaus and Montero) of biclique graphs that does not hold for square graphs.
We study properties of graphs in terms of ordering of the vertices of their cliques and neighbourhood (Theorem 2). That property lead to a forbidden structure () for mutually included biclique graphs. We need to find other properties of these graphs in order to better understand their square graphs.
We also show that the class of biclique graphs of bipartite graphs are exactly the square of a subclass of comparability graphs (IIC-comparability). These characterizations (partial in the case of -free graphs) do not lead to polynomial time recognition algorithms. However, it gives another tool to study the problem of recognizing biclique graphs and their properties.
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