Circularly compatible ones, D-circularity, and proper circular-arc bigraphs

06/02/2019 ∙ by Martín D. Safe, et al. ∙ 0

In 1969, Alan Tucker characterized proper circular-arc graphs as those graphs whose augmented adjacency matrices have the circularly compatible ones property. Moreover, he also found a polynomial-time algorithm for deciding whether any given augmented adjacency matrix has the circularly compatible ones property. These results allowed him to devise the first polynomial-time recognition algorithm for proper circular-arc graphs. However, as Tucker himself remarks, he did not solve the problems of finding a structure theorem and an efficient recognition algorithm for the circularly compatible ones property in arbitrary matrices (i.e., not restricted to augmented adjacency matrices only). In this work, we solve these problems. More precisely, we give a minimal forbidden submatrix characterization for the circularly compatible ones property in arbitrary matrices and a linear-time recognition algorithm for the same property. We derive these results from analogous ones for the related D-circular property. Interestingly, these results lead to a minimal forbidden induced subgraph characterization and a linear-time recognition algorithm for proper circular-arc bigraphs, solving a problem first posed by Basu, Das, Ghosh, and Sen [J. Graph Theory, 73(4):361--376, 2013]. Our findings generalize some known results about D-interval hypergraphs and proper interval bigraphs.

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1 Introduction

In 1969, Alan Tucker [38] introduced the linearly compatible ones property in connection with a characterization of proper interval graphs in terms of their augmented adjacency matrices due to Fred Roberts [26]. In order to state this characterization, we now give the necessary definitions.

Let be a linear order on some set . If , then the linear interval of with left endpoint and right endpoint , denoted , is the set . A linear interval of is either the empty set or for some such that . A sequence is monotone on if and .

All matrices in this work are binary; i.e., have only and entries. We will usually identify each row of a matrix with the set of columns of having a at row . For instance, we say a row is empty if it has no entries, while a row is contained in a row , denoted , if has a at each column where has a . A row of is trivial if it is either empty or the set of all columns of . We adopt analogous conventions for the columns.

A biorder of a matrix

is an ordered pair

such that and are linear orders of the rows and of the columns of , respectively. A matrix has the linearly compatible ones property [38, p. 43] if admits some biorder such that: (i) each row of is a linear interval of ; (ii) each column of is a linear interval of ; and (iii) if are all the nontrivial rows of in ascending order of and equals the linear interval for each , then the sequences and are monotone on .222The definitions of the linearly compatible ones property and the circularly compatible ones property used in this work are taken directly from Tucker’s PhD thesis [38] and should not be confused with the ones given in [34, 35, 36] in the setting of symmetric matrices. If so, is called a linearly compatible ones biorder of .

Proper interval graphs is a well-known class of intersection graphs. The intersection graph of a family of sets is a graph having one vertex for each set of the family and having an edge joining two different vertices if and only if the sets of the family corresponding to these two vertices have nonempty intersection. A proper interval graph [25] is the intersection graph of a family of intervals on a line no two of which are one a proper subset of the other. Proper interval graphs admit many different characterizations [5, 11, 16, 22, 24, 25, 26, 30, 39]. The result below is the aforementioned characterization by Roberts of proper interval graphs in terms of their augmented adjacency matrices. An augmented adjacency matrix of a graph is any matrix that arises from an adjacency matrix by adding ’s all along the main diagonal.

Theorem 1 ([26]).

A graph is a proper interval graph if and only if its augmented adjacency matrix has the linearly compatible ones property.

Many notions which turn out to be equivalent to the linearly compatible ones property were subsequently introduced by Moore [23] (-interval hypergraphs), Spinrad, Brandstädt, and Stewart [31] (adjacency and enclosure property), Sen and Sanyal [29] (monotone consecutive arrangements), and Lai and Wei [18] (forward-convex labelings). For instance, a matrix has the -interval property [23] if there is a linear order of its columns such that each row of is a linear interval of and the set difference is also a linear interval of for any two rows and of . Moore [23] characterized the -interval property by minimal forbidden induced submatrices. Spinrad, Brandstädt, and Stewart [31] gave the first linear-time recognition algorithm for the -interval property. A significantly simpler linear-time recognition algorithm was later proposed by Sprague [32]. More recently, Hell and Huang [15] proposed a linear-time recognition algorithm which, in addition, outputs a minimal forbidden submatrix of the input matrix whenever does not have the property.

Interestingly, the linearly compatible ones property also characterizes proper interval bigraphs. The bipartite intersection graph [12] of two families of sets and is a graph having a vertex for each element of and for each element of and such that a vertex corresponding to an element in is adjacent to a vertex corresponding to an element in if and only if these two elements have nonempty intersection. A proper interval bigraph [29] is the bipartite intersection graph of two families and of intervals on a line where neither nor contains two intervals such that one is a proper subset of the other. If so, is called a proper interval bimodel of the bipartite intersection graph of and . Proper interval bigraphs admit several different characterizations [3, 4, 6, 7, 14, 15, 20, 19, 33, 29, 40]. Interestingly, proper interval bigraphs are known to coincide with many other graph classes, including unit interval bigraphs [29], bipartite permutation graphs [33], bipartite asteroidal-triple-free graphs [10], bipartite co-comparability graphs [10], bipartite tolerance graphs [3], and the complement of two-clique circular-arc graphs [14]. The bipartite graph associated with a matrix has one vertex for each row and for each column of and the vertex corresponding to row is adjacent to the vertex corresponding to column if and only if the entry of is . The result below follows by combining results from the works of Sen and Sanyal [29], Lai and Wei [18], and Moore [23] (see Subsection 4.1).

Theorem 2 ([18, 23, 29]).

Proper interval bigraphs are precisely the bipartite graphs associated with matrices having the linearly compatible ones property.

The above theorem allows for translation back and forth between results about the linearly compatible ones property and results about proper interval bigraphs. In fact, the aforementioned linear-time recognition algorithms in [15, 31, 32] for the linearly compatible ones property were originally formulated as recognition algorithms for proper interval bigraphs (or, equivalently, bipartite permutation graphs).

Tucker [38] introduced the circularly compatible ones property in order to characterize proper circular-arc graphs. If is a linear order on some set and , then the circular interval of with left endpoint and right endpoint , denoted , is either if , or if . A circular interval of is either the empty set or for some . A sequence is circularly monotone on if and holds for all but at most one (where stands for ). A matrix has the circularly compatible ones property [38, p. 30] if admits some biorder such that: (i) each row of is a circular interval of ; (ii) each column of is a circular interval of ; and (iii) if are all the nontrivial rows of in ascending order of and for each , then the sequences and are circularly monotone on . If so, is called a circularly compatible ones biorder. A proper circular-arc graph [38, 35] is the intersection graph of a family of arcs on a circle such that no two arcs are one a proper subset of the other. Proper circular-arc graphs admit several different characterizations [13, 30, 38, 35, 37]. The result below shows that Theorem 1 extends to proper circular-arc graphs and, in fact, this was the motivation behind the introduction of the circularly compatible ones property by Tucker.

Theorem 3 ([38, p. 36]).

A graph is a proper circular-arc graph if and only if its augmented adjacency matrix has the circularly compatible ones property.

Based on the above characterization, Tucker was able to devise the first polynomial-time recognition algorithm for proper circular-arc graphs by reducing the problem to that of deciding whether any given augmented adjacency matrix has the circularly compatible ones property [38, Section 2.2]. However, as Tucker himself remarks [38, p. 46], he did not solve the problems of finding a structure theorem and an efficient recognition algorithm for the circularly compatible ones property in arbitrary matrices (i.e., not restricted to augmented adjacency matrices only). We solve these problems. In order to do so, we study the following circular variant of the -interval property. A matrix has the -circular property if there is a linear order of its columns such that each row of is a circular interval of and the set difference is also a circular interval of for any two rows and of . If so, is a -circular order of . Hypergraphs whose incidence matrices have the -circular property were studied by Köbler, Kuhnert, and Verbitsky [17] (where they are called tight circular-arc hypergraphs).

Proper circular-arc bigraphs are defined analogously to proper interval bigraphs as follows. A proper circular-arc bigraph [14] is the bipartite intersection graph of two families and of arcs on a circle where neither nor contains two arcs such that one is a proper subset of the other. If so, is called a proper circular-arc bimodel of the bipartite intersection graph of and . In [7], proper circular-arc bigraphs were characterized in terms of a pair of linear orders of their vertices. Basu et al. [1] proved an analogue of Theorem 2 for proper circular-arc bigraphs having a biadjacency matrix with no trivial rows, where the linearly compatible ones property is replaced with the -circular property. Combining their result with our findings about the circularly compatible ones property, we derive the following analogue of Theorem 2 for arbitrary proper circular-arc bigraphs by replacing the linearly compatible ones property with the circularly compatible ones property.

Theorem 4.

Proper circular-arc bigraphs are the bipartite graphs associated with matrices having the circularly compatible ones property.

Basu et al. [1] asked for an efficient recognition algorithm for proper circular-arc bigraphs; more recently, Das and Chakraborty [8] raised the same problem. We solve this problem by giving a linear-time algorithm for recognizing proper circular-arc bigraphs. Moreover, as a consequence of the above theorem and our minimal forbidden submatrix characterization of the circularly compatible ones property, we derive a minimal forbidden induced subgraph characterization for proper circular-arc bigraphs.

The main results of this work are a minimal forbidden submatrix characterization and a linear-time recognition algorithm for the -circular property for arbitrary matrices. Moreover, we show that an arbitrary matrix has the circularly compatible ones property if and only if both the matrix and its transpose have the -circular property. As a consequence, we derive a minimal forbidden submatrix characterization for the circularly compatible ones property together with a linear-time recognition algorithm (thus solving the aforementioned problems by Tucker [38]). Given the connection between proper circular-arc bigraphs and the circularly compatible ones property (Theorem 4), these results lead to a minimal forbidden induced subgraph characterization and a linear-time recognition algorithm for proper circular-arc bigraphs (thus solving the problem first posed by Basu et al. [1]). Our recognition algorithms for matrix properties output either the linear order(s) required by the definition of the property or a minimal forbidden submatrix. Similarly, our recognition algorithms for graph classes either produce a bimodel as required by the definition of the class or a minimal forbidden induced subgraph.

This work is organized as follows. In Section 2, we give basic definitions and notation and state some previous results about the consecutive-ones and the circular-ones properties. In Section 3, we give a minimal forbidden submatrix characterization and a linear-time recognition algorithm for the -circular property and discuss their connection with some known results about the -interval property. In Section 4, we argue that the linearly compatible ones property is equivalent to the -circular property and give a minimal forbidden submatrix characterization and a linear-time recognition algorithm for the circularly compatible ones property. In Section 5, we derive a minimal forbidden induced subgraph characterization and a linear-time recognition algorithm for proper circular-arc bigraphs. Some of the proofs of the more technical results are given in Appendix A.

2 Definitions and preliminaries

For each positive integer , we denote by the set ; if , denotes the empty set. We also denote by the set endowed with the natural order. In the same vein, if , we write to denote the circular interval with left endpoint and right endpoint with respect to the natural order of . By we denote the identity function with domain . Let be a linear order on some finite set . If , we write to mean and ; this convention also applies to linear orders denoted by with some subscript and/or superscript (e.g., , , , etc.). A set is properly contained in or is a proper subset of a set if is a subset of and . Two sets and are incomparable if none of them is a subset of the other.

Sequences

Let be a sequence of length . We call the shift of to the sequence and the reversal of to the sequence . We denote the length of any sequence by . If is a sequence and , we say that occurs circularly in at position if and where subindices are modulo . If , we may simply say that occurs in at position .

If is binary (i.e., each is either or ), we define the complement of , denoted by , as the sequence that arises from by interchanging ’s with ’s. A binary bracelet [28] is a lexicographically smallest element in an equivalence class of binary sequences under shifts and reversals.

A sequence is senary if each of its element is , , , , , or . We define the complement of a senary sequence , denoted by , as the sequence that arises from by interchanging ’s with ’s, ’s with ’s, and ’s with ’s.

Matrices

Let and be matrices. We say that contains as a configuration if some submatrix of equals up to permutations of rows and of columns. We say that and represent the same configuration if and are equal up to permutations of rows and of columns; otherwise, we say that and represent different configurations. We denote by the matrix that arises from by adding one last column consisting entirely of ’s. We denote the transpose of by .

Let be a matrix. We assume that the rows and columns of are labeled from to and from to , respectively, as usual. By complementing row of we mean replacing, in row , all entries by ’s and all entries by ’s. The complement of , denoted , is the matrix arising from by replacing all entries by ’s and all entries by ’s. If is a binary sequence of length , we denote by the matrix that arises from by complementing those rows such that . A row map of is an injective function for some positive integer . A column map of is an injective function for some positive integer . If is a row map of and is a column map of , we denote by the matrix such that, for each , its -entry is the -entry of . We also write to mean . Notice that contains as a configuration if and only if there is a row map and column map of such that . If, in addition, is a binary sequence whose length equals the number of rows of , then . It is also clear that, if and are a row map and a column map of and and are a row map and a column map of , then and are a row map and a column map of , and . If is a positive integer and are pairwise different positive integers, we denote by the injective function with domain that transforms into for each . Hence, if , then denotes the matrix whose only row equals row of .

The canonical order of the rows of is the linear order of the rows of as they occur from top to bottom. Similarly, the canonical order of the columns of is the linear order of the columns of as they occur from left to right. The canonical biorder of is the biorder where and are the canonical orders of the rows and of the columns of , respectively.

Graphs

All graphs in this work are simple; i.e., finite, undirected, and with no loops and no multiple edges. If is a graph and is some subset of its vertex set, the subgraph of induced by is the graph having as vertex set and whose edges are the edges of having both endpoints in . An induced subgraph of some graph is the subgraph of induced by some subset of its vertex set. An isolated vertex of a graph is a vertex of the graph adjacent to no vertex in the graph.

A stable set of a graph is a set of pairwise nonadjacent vertices. A bipartition of a graph is a partition of its vertex set into two (possibly empty) stable sets. A graph is bipartite if it admits a bipartition. Let be a bipartite graph and let be a bipartition of . A biadjacency matrix of with respect to and has one row for each vertex in and one column for each vertex in and, for each and each , the entry in the intersection of the row corresponding to and the column corresponding to is if and only if is an edge of . The bipartite complement of with respect to is the bipartite graph with bipartition such that, for each and each , is adjacent to in if and only if is nonadjacent to in . A bipartite complement of is the bipartite complement of with respect to some bipartition of .

Algorithms

If is a matrix, we denote by the sum of the number of rows, the number of columns, and the number of ones of . We say that an algorithm taking a matrix as input is linear-time if it runs in time. In time and space bounds of algorithms taking a graph as input, we denote by and the number of vertices and edges of the input graph. We say that an algorithm taking a graph as input is linear-time if it runs in time. We assume that input matrices are represented by lists of rows, where each row is represented by a list of the columns having a in the row. We assume input graphs are represented by adjacency lists. This way, matrices and graphs are represented in and space, respectively.

Consecutive-ones property and circular-ones property

A matrix has the consecutive-ones property for rows [9] (resp. circular-ones property for rows [35]) if there is a linear order of the columns of such that each row of is a linear interval (resp. a circular interval) of . If so, is called a consecutive-ones order (resp. a circular-ones order) of . The consecutive-ones property for columns (resp. circular-ones property for columns) is defined analogously by reversing the roles of rows and columns. If no mention is made to rows or columns, we mean the corresponding property for the rows. If the canonical order of the columns of some matrix is a circular-ones order of , we say that is a circular-ones matrix.

Booth and Lueker [2] gave linear-time recognition algorithms for both the consecutive-ones property and the circular-ones property. (In the theorem below, denotes the matrix consisting of the first rows of , and is an instance of the notation introduced earlier in this section.)

Theorem 5 ([2]).

There is a linear-time algorithm that, given any matrix , outputs either a consecutive-ones order (resp. a circular-ones order) of or the least positive integer such that does not have the consecutive-ones (resp. circular-ones) property.

[] []   []   []

Figure 1: for each , where omitted entries are ’s
Figure 2:
Figure 3:
Figure 4: Some Tucker matrices, where denotes the number of rows

[]

[]

Figure 5:

[]

Figure 6:

[]

Figure 7:

[]

Figure 8:

[]

Figure 9:

[]

Figure 10:

[]

Figure 11:

[]

Figure 12:
Figure 13: Some small matrices

Tucker [36] characterized the consecutive-ones property by a minimal set of forbidden submatrices, known as Tucker matrices. The matrices for each , , and , displayed in Figure 4, are some of the Tucker matrices. In [27], we gave an analogous characterization for the circular-ones property. The corresponding set of forbidden submatrices is

where and denote and , and, for each , is the set of all binary bracelets of length . Notice that and are binary bracelets of length but do not belong to . A matrix is a minimal forbidden submatrix for the circular-ones property if is the only submatrix of not having the circular-ones property.

Theorem 6 ([27]).

A matrix has the circular-ones property if and only if contains no matrix in the set as a configuration. Moreover, there is a linear-time algorithm that, given any matrix not having the circular-ones property, outputs a matrix in contained in as a configuration. In addition, every matrix in is a minimal forbidden submatrix for the circular-ones property. Hence, for each and each binary sequence whose length equals the number of rows of , represents the same configuration as some matrix in .

3 -circular property

Figure 13 introduces the matrices , , , , , , , and needed in what follows. (Notice that .) The main aim of this section is to give a characterization by minimal forbidden submatrices and a linear-time recognition algorithm for the -circular property. The corresponding set of minimal forbidden submatrices is

where

All the matrices in are displayed explicitly later in Figure 29 (see Subsection 3.5). Notice that each of , , , , and represents the same configuration as its complement. Hence, for each matrix in , there is some in such that represents the same configuration as .

This section is organized as follows. In Subsection 3.1, we discuss the connection between the -circular property and the circular-ones property. In Subsections 3.2, 3.3, and 3.4 some auxiliary matrices are shown to contain as a configuration some matrix in or in ; these technical results are crucial for the proof of the minimal forbidden submatrix characterization for the -circular property given in Subsection 3.5. In Subsection 3.6, we give our linear-time recognition algorithm for the -circular property. Finally, in Subsection 3.7, we study the connection between the results about the -circular property obtained along this section and some known results for the -interval property.

3.1 Connection with the circular-ones property

If is a matrix, we denote by a matrix that arises from by adding rows at the bottom as follows: for each nontrivial rows and such that is properly contained in , add a row equal to the set difference . (This operator for matrices is intimately related but slightly different from the operator defined in [23] and the operator defined in [17] for hypergraphs .) As arises from by adding rows, we will usually regard the linear orders of the columns of as linear orders of the columns of and vice versa. The following fact is immediate consequence of the definitions.

Lemma 7.

A matrix has the -circular property if and only if has the circular-ones property.

Proof.

By definition, each -circular order of is a circular-ones order of . Hence, if has the -circular property, then has the circular-ones property. For the converse, suppose that has the circular-ones property. Thus, there is some circular-ones order of . Let and be two rows of . As and are rows of , then and are circular intervals of . Hence, is also a circular interval of unless, perhaps, when is properly contained in . If or is trivial, then is trivial, , or the complement of , all of which are circular intervals of . Thus, we assume, without loss of generality, that and are nontrivial. Therefore, if is properly contained in , then is a circular interval of because is a row of . We conclude that in all cases is a circular interval of . Thus, by definition, is a -circular order of . This proves that, whenever has the circular-ones property, has the -circular property. The proof of the lemma is complete.∎

By virtue of the above lemma and the fact that complementing some rows of a matrix preserves the circular-ones property, the following result implies that has the -circular property if and only if has the -circular property.

Lemma 8.

If is a matrix, then arises from by complementing its first rows and permuting the remaining rows.

Proof.

Let be the matrix that arises from by complementing its first rows. By definition of , the first rows of coincide with the first rows of . Let . Notice that is properly contained in if and only if is properly contained in . Moreover, . We conclude that the rows of and are the same up to permutation. This completes the proof of the lemma.∎

As none of the matrices in has the circular-ones property (see Theorem 6), Lemma 7 together with the result below shows that none of the matrices in has the -circular property.

Lemma 9.

If , then contains some matrix in as a configuration.

Proof.

If is or for some , then the lemma holds immediately because contains as a configuration and . Moreover, the following assertions can be verified by inspection: (i) if , then contains as a configuration; (ii) if , then contains as a configuration; (iii) if , then contains as a configuration; (iv) if , then contains as a configuration; (v) if , then contains as a configuration. Since , , , , and belong to , the proof of the lemma is complete.∎

Our next result shows that each matrix in contains some matrix in as a configuration.

Lemma 10.

If , then contains some matrix in as a configuration.

Proof.

If is or , then is or , respectively. If is or , then is or , respectively. Thus, it only remains to consider the case where for some and some binary sequence such that . If consists entirely of ’s or entirely of ’s, then coincides with or , both of which belong to . Hence, we assume, without loss of generality, that is nonconstant (i.e, contains at least a and at least a ) and, necessarily, .

Suppose first that for some (where subindices are modulo ). Thus, some sequence of length such that occurs circularly in at position . If we let and (where sums involving are modulo ), then and

On the one hand, if , then is , , , or and, consequently, , , , or , respectively. On the other hand, if , then, since and each of , , and represents the same configuration as its complement, the matrix represents the same configuration as , , or .

Suppose now that for each (where subindices are modulo ). Since is a nonconstant bracelet, its prefix of length must be , , or . Thus, is a multiple of and is the concatenation of copies of that prefix. As a consequence, some sequence occurs circularly in at position for some . If we let and (where the sums involving are modulo ), then ,

and, consequently, equals or depending whether is or , respectively. This completes the proof of the lemma.∎

3.2 Matrices and

We will associate with each senary sequence of length at least a matrix denoted . As a preliminary result for the proof of Theorem 15, we need to prove Lemma 11 below, which asserts that, for almost all senary sequences of length at least , contains some matrix in as a configuration. We now introduce the necessary definitions.

For each and each , we define the following matrices, where in all the cases should be understood modulo :

  • is the matrix whose only row has ’s at columns and and ’s at the remaining ones;

  • is the complement of ;

  • is the matrix whose first row has a at column and ’s at the remaining columns and whose second row has ’s at columns , , and and ’s at the remaining columns;

  • is the complement of ;

  • is the matrix whose first row has a at column and ’s at the remaining columns and whose second row has a at column and ’s at the remaining columns;

  • is the complement of .

Given a senary sequence of length for some , we denote by the matrix having columns and whose rows are those of , followed by those of , followed by those of , …, followed by those of . For instance,

Lemma 11.

Let be a senary sequence of length such that . If , suppose additionally that neither nor occurs in . Then, contains some matrix in as a configuration.

The proof of Lemma 11 is given in Section A.1 of the appendix.

3.3 Matrices and

We will now associate with some senary sequences of length , a corresponding matrix and we will show that, for certain such sequences , contains some matrix in as a configuration (see Lemma 12).

We first define, for each , the following matrices:

  • whose only rows coincides with row of ;

  • is the complement of .

For each , we define the following matrices, where sums involving are modulo :

  • is the matrix whose first row coincides with the complement of row of and the second row coincides with row of ;

  • is the complement of .

We need two sporadic matrices:

  • ;

  • is the complement of .

For each senary sequence of length such that and , we define as the matrix having six columns and whose rows are those of , followed those of , followed by those of , followed by those of . For instance,

Lemma 12.

If is a senary sequence such that and , then contains some matrix in as a configuration.

The proof of Lemma 12 is given in Section A.2 of the appendix.

3.4 Matrices and

For each binary sequence of length and each , we define as the matrix that arises from by adding a fifth row having ’s in columns and , and a sixth row having ’s in columns and and such that the entries at each of the columns , , , and (where additions are modulo ) coincide in the fifth and sixth rows and are equal to , , , and , respectively. For instance,

For each binary sequence of length , we define

Lemma 13.

Let be a binary sequence of length and let . If , then contains as a configuration some matrix in