Linear Algebra and its Applications 436 (2012) 3239–3246
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Boolean rank of upset tournament matrices David E. Brown a,∗ , Scott Roy a , J. Richard Lundgren b , Daluss J. Siewert c a b c
Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA Department of Mathematical Sciences, University of Colorado Denver, Denver, CO 80217, USA Department of Mathematics, Black Hills State University, Spearfish, SD 57799, USA
ARTICLE INFO
ABSTRACT
Article history: Received 8 September 2011 Accepted 4 November 2011 Available online 3 December 2011
The Boolean rank of an m × n (0, 1)-matrix M is the minimum k for which matrices A and B exist with M = AB, A is m × k, B is k × n, and Boolean arithmetic is used. The intersection number of a directed graph D is the minimum cardinality of a finite set S for which each vertex v of D can be represented by an ordered pair (Sv , Tv ) of subsets of S such that there is an arc from vertex u to vertex v in D if and only if Su ∩ Tv = Ø. The intersection number of a digraph is equal to the Boolean rank of its adjacency matrix. Using this fact, we show that the intersection number of an upset tournament, equivalently, the Boolean rank of its adjacency matrix, is equal to the number of maximal subpaths of certain types in its upset path. © 2011 Elsevier Inc. All rights reserved.
Submitted by R.A. Brualdi AMS classification: 15A23 12B34 05C20 05C62 05C70 Keywords: Boolean rank Intersection number Tournament Tournament matrix
1. Introduction The problem of determining the factor rank of (0, 1)-matrices with respect to certain semirings has applications which span many fields and has received significant attention over the past 30 years. For instance, the papers [2,6,14] by Beasley, de Caen, Kirkland, Maybee, Pullman, and Shader give results about the real ranks and semiring ranks of some specialized matrices, such as tournament matrices. The thesis of Shader [19] gives a complete determination of the nonnegative integer rank of upset tournament matrices. Our results complement Shader’s in the sense that we determine another semiring rank, the Boolean rank, for upset tournament matrices. Many results about the Boolean ranks ∗ Corresponding author. E-mail address:
[email protected] (D.E. Brown). 0024-3795/$ - see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2011.11.003
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of varied classes of matrices can be found in the papers [1,3,7,8,10–13,20] by Beasley, de Caen, Factor, Kohler, Darby, Gregory, Hefner, Jones, Lundgren, Pullman, and Watts. The facts we use about the nature of upset tournaments come from the papers [16] by Poet and Shader, and [5] by Brualdi and Li. Our results are helped by the perspective of representing a directed graph by set intersections. The problem of representing an undirected graph by set intersections was first introduced in [9] by Erdos, ˝ Goodman, and Pósa. A graph G is represented by a set S if the vertices of G correspond to subsets of S. In [9] the minimum cardinality of S was shown to be the minimum number of complete subgraphs which cover the edges of the graph. We will use the analogous notion of representing a directed graph via set intersections of ordered pairs of sets of some superset, as introduced in [18]. Specifically, each vertex v of digraph D corresponds to an ordered pair (Sv , Tv ), where Sv , Tv ⊆ S and there is an arc from u to v in D if and only if Su ∩ Tv = Ø. We show that the minimum size of the superset S is equal to the Boolean rank of the digraph’s adjacency matrix, which via a result in [12], is equal to the minimum number of directed bicliques which cover the arcs of the digraph. 2. Preliminaries one In this section we give the connection between Boolean rank and intersection number and develop most of the notation we will use. Let D = (V , A) denote a directed graph with no multiple arcs (henceforth simple digraph or simply digraph) with vertex set V or V (D), and arc set A or A(D). If (u, v) ∈ A we may write u → v in D, or u beats v, and may refer to u as the tail of arc (u, v) and v as its head. The number of vertices a vertex v beats, that is |{u ∈ V (D) : v → u in D}|, is denoted by d+ (v). We say that D has an intersection representation if there is a set S such that to each vertex v of D we can name an ordered pair of subsets (Sv , Tv ) of S with u → v in D if and only if Su ∩ Tv = Ø. This model was first introduced in [18] and has been studied heavily in the case where S is the set of intervals of the real line – that is, the case where D is an interval digraph. For more background on interval digraphs see, for example, the papers [4,15,17,18,21] by Brown, Busch, Lundgren, Müller, Sanyal, Sen, Das, and West. Every digraph has an intersection representation where S is finite. To wit, let D be a digraph with A(D) labeled somehow; put S equal to the set of labels of A, for each v ∈ V (D), put Sv equal to the set of labels of arcs for which v is the tail, and put Tv equal to the set of labels of arcs for which v is the head. Clearly, Su ∩ Tv = Ø if and only if u → v in D. The minimum cardinality of S for which D has an intersection representation is the intersection number of D and we denote it by in(D). For undirected graphs the corresponding parameter was introduced in [9] and shown to be equal to the minimum number of complete subgraphs which cover the edges of the graph. We will show that in(D) is equal to the minimum number of directed bicliques which cover the arcs of D, and hence the Boolean rank of D’s adjacency matrix. A directed biclique of digraph D is a subdigraph of D on a pair of sets X ⊆ V (D) and Y ⊆ V (D) in which every vertex in X beats each vertex in Y ; if D has loops, X and Y are possibly not disjoint. We denote the minimum number of directed bicliques needed to cover the arcs of digraph D by bc (D). The Boolean rank of a (0, 1)- m × n matrix M, denoted rB (M ), is the minimum k for which there is a factorization M = AB using Boolean arithmetic (1 + 1 = 1) where A is m × k and B is k × n. Equivalently, rB (M ) is the minimum number of rank-one matrices whose Boolean sum is M. In [12] it was shown rB (M ) = bc (D), where M is D’s adjacency matrix; this result follows, basically, because a directed biclique corresponds to a rank-one matrix. Suppose M is the adjacency matrix of digraph D = ({v1 , v2 , . . . , vn }, A); that is, M is a (0, 1)matrix, and if M = (mij ), then mij = 1 if and only if vi → vj in D. Any intersection representation of D via S yields a Boolean factorization of M into M = ST, where S is a (0, 1)- (n × |S|)-matrix and T is a (0, 1)- (|S| × n)-matrix via the following construction. Suppose D has an intersection representation using S = {1, 2, . . . , k}. Put S equal to the (0, 1)-matrix S = (sij ) with sij = 1 if and only if the set Svi contains element j of S; put T equal to the (0, 1)-matrix T = (tij ) with tij = 1 if and only if element i of S is contained in set Tvj . Hence mij = kr=1 sir trj = 1 if and only if sir trj = 1 for some r ∈ {1, 2, . . . , k} if and only if Svi ∩ Tvj = Ø. This shows rB (M ) in(D), but the reverse inequality also follows: take a Boolean factorization M = ST, use the labels on the columns of S and rows of T as S, put Svi = {r : sir = 1} and Tvj = {r : tri = 1}. Now Svi ∩ Tvj = Ø if and only if mij = 1.
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From the arguments in the previous paragraph, we see that the intersection number of a digraph is equal to the Boolean rank of its adjacency matrix. Henceforth, we will denote the adjacency matrix of a digraph D by [D]. Lemma 2.1. Let D be a simple digraph. Then rB ([D])
= in(D).
3. Preliminaries two Our main result is the determination of the biclique cover number of any upset tournament. This result will be developed through the perspective afforded by a scheme for coloring the arcs which corresponds to an intersection representation of the tournament. In this section we develop this coloring scheme and the corresponding terminology. Suppose D is a digraph and that S = {c1 , c2 , . . . , ck } is the set used in an intersection representation {(Su , Tv ) | (u, v) ∈ A(D)}. Think of the elements of S as colors and color each arc u → v of D with all colors in Su ∩ Tv . Note that an arc may be colored with many colors. We may refer to Sv as the color-outset of v and Tv as the color-inset of v. The coloring scheme yields the relationship u → v if and only if the color outset of u intersects the color inset of v. We will capitalize on the fact that the number of colors required is equal to the intersection number of D and hence to the Boolean rank of [D]. For example, suppose D is a tournament, that is, D is an orientation of a complete graph. Hence D has no arcs of the form v → v, and so the color of any arc for which v is the tail must be distinct from the color of any arc for which v is the head – otherwise Sv ∩ Tv = Ø and v → v. Also, if D is a tournament and v1 , v2 , v3 , v4 , in that order, form a path in D, then the arcs vi → vi+1 (1 i 3) must be colored distinctly, for if not, then Sv3 ∩ Tv2 = Ø and we have both v2 → v3 and v3 → v2 . We call a tournament an n-tournament if it is a tournament whose vertex set has size n. By results in [5,16] we may regard an upset n-tournament on V = {v1 , v2 , . . . , vn } as one obtained from the transitive n-tournament on V , with vi → vj if i>j, by reversing arcs so that a unique (by [16]) path from v1 to vn is formed which we will call the upset path. Note that we may assume, relabeling if necessary, that v1 → v2 and vn−1 → vn . An upset n-tournament T is in standard form if the vertices of T are labeled v1 , v2 , . . . , vn with d+ (v1 ) = 1, d+ (vn ) = n − 2, d+ (vi ) = i − 1 for 2 i n − 1, and the arcs v1 → v2 and vn−1 → vn are in T. We always assume an upset n-tournament is in standard form. We now define some terms we will use to speak about substructures of upset tournaments. The complement upset path in upset n-tournament T is the path from and including vn to and including v1 which consists of all vertices in {vn−2 , . . . , v3 } which are not part of the upset path. Note that the upset path union the complement upset path is a Hamiltonian cycle. The arc (vi , vj ) in the upset path is contained in the arc (vl , vk ) of the complement upset path if and only if l j and i k; we may also say (vl , vk ) contains (vi , vj ). A consecutive path of length l in the upset path of upset n-tournament T is a path of the form vi → vi+1 → vi+2 → · · · → vi+l−1 → vi+l , where l > 0. We classify consecutive paths in the following four ways.
• • • •
A beginning consecutive path is a consecutive path with i = 1 and i + l = n. An ending consecutive path is a consecutive path with i = 1 and i + l = n. A complete consecutive path is a consecutive path with i = 1 and i + l = n. An internal consecutive path is a consecutive path with i = 1 and i + l = n.
A reducing consecutive path in an upset tournament T is a consecutive path in the upset path U of T which is not contained in a longer consecutive path in U and which is of one of the following types: beginning of length l 3; ending of length l 3; complete of length l 5; or, internal of length l 1. 4. Lower bound for the Boolean rank of an upset tournament matrix Suppose T is an upset n-tournament, and [T ] = AB is a Boolean factorization. Recall from the previous section that a coloring of the arcs of T is obtained through this factorization. In this section
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we employ the arc-coloring scheme to establish a lower bound for the biclique cover number of any upset tournament. Equivalently, we determine a lower bound on the size of set S through which T has an intersection representation. Recall the color(s) of an arc u → v are all elements of Su ∩ Tv . In Lemmas 4.1–4.6, T is an upset n-tournament. Note that if vi → vj and vl → vk are arcs in D that share a color in S, then Svi ∩ Tvj ∩ Svl ∩ Tvk = Ø and so vi → vk and vl → vj must be arcs in D as well. Lemma 4.1. No two arcs in the upset path of upset tournament T share a color. Proof. Suppose vi → vj and vl → vk are arcs of the upset path of T that share a color and i vi → vk , so vi is the tail of another upward arc of T, a contradiction.
< k. Then
Lemma 4.2. No two arcs in the complement upset path of upset tournament T share a color. Proof. Suppose vi → vj and vl → vk are two arcs in the complement upset path where i > j l > k. If the arcs share a color, then vl → vj . But the vertices are on the complement upset path, so l > j, a contradiction. Lemma 4.3. If the arc vi → vj in the upset path of T shares a color with vl path of T, then vl → vk contains vi → vj .
→ vk in the complement upset
Proof. Suppose vi → vj is an arc of the upset path of T and vl → vk is an arc of the complement upset path of T. If the arcs share a color, then vi → vk and vl → vj are arcs of T. These arcs cannot be arcs of the upset path since vi → vj is the only upward arc with tail vi and the only upward arc with head vj . Thus, i > k and l > j, so vi → vj is contained in vl → vk . Lemma 4.4. If an arc vl → vk in the complement upset path of T contains an arc vi path, then j = i + 1 so vi → vj belongs to a consecutive path.
→ vj in the upset
Proof. Assume vl → vk contains vi → vj , and that vi → vj is not part of a consecutive path. Then j > i + 1 and vi+1 belongs to the complement upset path, but since l j > i + 1 > i k, vl → vk cannot be an arc of the complement upset path. Lemma 4.5. If an arc vl → vk of the complement upset path of T shares a color with vi path, then vl → vk contains a consecutive path to which vi → vj belongs.
→ vj in the upset
Proof. This result follows from Lemmas 4.3 and 4.4. Lemma 4.6. No arc in a beginning or ending consecutive path of length l 2 in the upset path can share a color with an arc in the complement upset path. Also, no arc in a complete consecutive path of length l 4 can share a color with an arc in the complement upset path. Proof. Apply Lemma 4.5 and the fact that arcs along a path of length three must be distinct in color. We are now ready to state and prove the main result of this section. Theorem 4.7. If T is an upset n-tournament, then the Boolean rank of [T ] is greater than or equal to n − q, where q is the number of reducing consecutive paths in the upset path of T. Proof. Suppose T is an upset n-tournament and consider the unique Hamiltonian cycle H consisting of the upset path and complement upset path. We show that the arcs of H can be colored with no fewer than n − q colors, where q is the number of reducing consecutive paths. From Lemmas 4.1 and 4.2 each arc in the upset path is distinctly colored, and each arc in the complement upset path is distinctly colored. Now, by Lemmas 4.5 and 4.6 each arc in the complement
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Fig. 1. Each reducing consecutive path is associated with a subpath of the upset path of one of the types shown.
upset path which has the same color as an arc in the upset path must be an arc that contains a reducing consecutive path. Since each of the q reducing consecutive paths is contained in precisely one arc of the complement upset path, at most q of the arcs in the complement upset path have the same color as an arc of the upset path. Thus, at least n − q of the n arcs of the Hamiltonian cycle must have distinct colors.
5. Boolean rank of an upset tournament matrix In this section we show that equality holds in Theorem 4.7. Suppose T is an upset n-tournament on vertices v1 , v2 , . . . , vn (labeled as discussed in Section 3) and that T has q reducing consecutive paths in its upset path U. Note that each of the q reducing consecutive paths is associated with one of the six types B3 , B4 , B, N1 , N2 , N shown in Fig. 1. Using these six types, we will see below that there are q distinct columns in [T ] (one for each consecutive path and associated type) each of which is a Boolean sum of a selection of succeeding columns in [T ]. This then implies that rB ([T ]) n − q, giving equality in Theorem 4.7. If B3 is a subpath of U, then
1 2 3 4
[T ] = 5
j
⎛
1 2 3 4 5
0 ⎜ ⎜0 ⎜ ⎜ ⎜1 ⎜ ⎜ ⎜1 ⎜ ⎜ ⎜1 ⎜ ⎜ ⎜ .. ⎜. ⎜ ⎜ ⎜1 ⎜ ⎝
1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 1 1 0
.. .. .. .. . . . .
j
··· ··· ··· ··· ··· .. .
0 0 0 1 0
.. .
··· ··· ··· ··· ···
··· 0 ··· .. . . .. .. .. .. .. . . . . . . . 1 1 0 1
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
and column 1 is the Boolean sum of columns 4, 5, and j.
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If B4 is a subpath of U, then
1
[T ] =
⎛
1 2 3 4 5 6
j
··· ··· ··· ··· ··· ··· .. . ···
0 1 0 0 0 0
⎜ 2 ⎜ ⎜0 ⎜ 3 ⎜1 ⎜ 4 ⎜ ⎜1 ⎜ 5 ⎜ ⎜1 ⎜ 6 ⎜1 ⎜ ⎜. ⎜. ⎜. ⎜ j ⎜ ⎜1 ⎝
0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 1 1 0 0 0 1 1 1 1 0
.. .. .. .. .. . . . . .
1 1 1 0 1
.. .. .. .. .. .. . . . . . .
0 0 0 0 1 0
.. .
··· ··· ··· ··· ··· ···
··· .. . . . .
0
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
and column 1 is the Boolean sum of columns 4, 5, and j. If B is a subpath of U, then ⎛ 1 2 3
[T ] =
4 5 6
1 2 3 4 5 6
0 ⎜ ⎜0 ⎜ ⎜ ⎜1 ⎜ ⎜ ⎜1 ⎜ ⎜ ⎜1 ⎜ ⎜1 ⎜ ⎝.
··· ··· ··· ··· ··· ··· .. .
1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 1 1 0 0 1 1 1 1 0 0
.. ... ... ... ... ...
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
and column 1 is equal to the Boolean sum of columns 4, 5, and 6. If N1 is a subpath of U, then ⎛
k
. . .. ⎜ . . ⎜ k ⎜ ⎜· · · 0 ⎜ .. ⎜ ⎜ . ⎜ ⎜ ⎜· · · 1 ⎜ i ⎜ ⎜· · · 1 [T ] = ⎜ ⎜· · · 1 ⎜ ⎜ ⎜· · · 1 ⎜ ⎜ .. ⎜ . ⎜ ⎜ j ⎜ ⎜· · · 1 ⎝ .. .
i
··· .. . ··· ··· ··· ···
j
.. .. .. .. . . . .
.. .
0 1 0 0
··· 0 .. . ··· 0 ··· 0 ··· 1 ··· 0 . . .. . . ··· 0 .. .
.. .. .. .. . . . .
0 0 0 0 1 0 1 0 1 0 0 0 1 1 1 0
.. .. .. .. . . . .
··· 1 .. .
1 0 1
.. .. .. . . .
⎞ ⎟
⎟ ··· ⎟ ⎟
··· ··· ··· ··· ··· .. .
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
and column (i − 1) is the Boolean sum of columns (i + 1), (i + 2), and j.
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If N2 is a subpath of U, then ⎛
[T ] =
k
.
⎜ .. ⎜ k ⎜ ⎜· · · ⎜ ⎜ ⎜ ⎜ ⎜ ⎜· · · ⎜ ⎜ i ⎜· · · ⎜ ⎜· · · ⎜ ⎜ ⎜· · · ⎜ ⎜ ⎜· · · ⎜ ⎜ ⎜ ⎜ ⎜ j ⎜ ⎜· · · ⎝
i
j
.. .
.. .. .. .. .. . . . . .
.. .
··· .. . . . . 1 ··· 0 ··· 1 ··· 1 ··· 1 ··· .. . 1 ··· .. .
0 1 0 0 0
··· 0 .. . ··· 0 ··· 0 ··· 0 ··· 1 ··· 0 . . .. . . ··· 0 .. .
0
.. .. .. .. .. . . . . .
0 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 1 0 0 0 1 1 1 1 0
.. .. .. .. .. . . . . .
1 1 1 0 1
.. .. .. .. .. . . . . .
⎞
··· ··· ··· ··· ··· ··· ··· .. .
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
and column (i − 1) is the Boolean sum of columns (i + 1), (i + 2), and j. If N is a subpath of U, then ⎛
k
. . .. ⎜ . . ⎜ k ⎜ ⎜· · · 0 ⎜ .. ⎜ ⎜ . ⎜ ⎜ ⎜· · · 1 ⎜ [T ] = i ⎜ ⎜· · · 0 ⎜ ⎜· · · 1 ⎜ ⎜ ⎜· · · 1 ⎜ ⎜ ⎜· · · 1 ⎝ .. .
i
⎞
.. .. .. .. .. . . . . . ··· .. . ··· ··· ··· ··· ···
0 1 0 0 0
···
0 0 0 0 0
··· ··· ··· ··· ··· .. .
.. .. .. .. .. . . . . .
1 0 1 0 0 1 0 0 1 0 1 1 0 0 1 1 1 1 0 0
.. .. .. .. .. . . . . .
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
and column (i − 1) is equal to the Boolean sum of columns (i + 1), (i + 2), and (i + 3). We have now established our main result. Theorem 5.1. If T is an upset n-tournament, then in(T ) = bc (T ) number of reducing consecutive paths in the upset path of T.
= rB ([T ]) = n − q, where q is the
References [1] [2] [3] [4] [5] [6] [7]
L.B. Beasley, A.E. Guterman, Rank inequalities over semirings, J. Korean Math. Soc. 42 (2005) 223–241. L.B. Beasley, S.J. Kirkland, B.L. Shader, Rank comparisons, Linear Algebra Appl. 221 (1995) 171–188. L.B. Beasley, N.J. Pullman, Semiring rank versus column rank, Linear Algebra Appl. 101 (1988) 33–48. D.E. Brown, A.H. Busch, J.R. Lundgren, Interval tournaments, J. Graph Theory 56 (2007) 72–81. R. Brualdi, Q. Li, Upsets in round robin tournaments, J. Comb. Theory B 35 (1983) 62–77. D. de Caen, The ranks of tournament matrices, Am. Math. Mon. 98 (1991) 62–77. D. de Caen, D. Gregory, N. Pullman, The Boolean rank of zero one matrices, in: Proceedings of the Third Caribbean Conference on Combinatorics and Computing, Barbados, 1981, pp. 169–173. [8] D. de Caen, D. Gregory, N. Pullman, The Boolean rank of zero one matrices, II, in: Proceedings of the Fifth Caribbean Conference on Combinatorics and Computing, Barbados, 1988, pp. 120–126. [9] P. Erdös, A.W. Goodman, L. Pósa, The representation of a graph by set intersection, Can. J. Math. 18 (1966) 106–112.
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D.E. Brown et al. / Linear Algebra and its Applications 436 (2012) 3239–3246
[10] K.A.S. Factor, R.M. Kohler, J.M. Darby, Local out-tournaments with upset tournament strong components: I. Full and equal {0, 1}-matrix ranks, Ars Combinatoria 95 (2010) 3–17. [11] D.A. Gregory, N.J. Pullman, Semiring rank: Boolean rank and nonnegative rank factorizations, J. Comb. Inf. Syst. Sci. 8 (1983) 223–233. [12] D.J. Gregory, K.F. Jones, J.R. Lundgren, N.J. Pullman, Biclique coverings of regular bigraphs and minimum semiring ranks of regular matrices, J. Comb. Theory B 51 (1991) 73–89. [13] K.A.S. Hefner, J.R. Lundgren, Minimum matrix ranks of k-regular (0, 1)-matrices, Linear Algebra Appl. 133 (1990) 43–52. [14] J.S. Maybee, N.J. Pullman, Tournament matrices and their generalizations, Linear Multilinear Algebra 28 (1990) 58–70. [15] H. Müller, Recognizing interval digraphs and interval bigraphs in polynomial time, Discrete Appl. Math. 78 (1997) 189–205. [16] J.L. Poet, B.L. Shader, Score certificate numbers for upset tournaments, Discrete Appl. Math. 103 (2000) 177–189. [17] B.K. Sanyal, M.K. Sen, Indifference digraphs: a generalization of indifference graphs and semiorders, SIAM J. Discrete Math. 7 (1994) 157–165. [18] M. Sen, S. Das, A.B. Roy, D.B. West, Interval digraphs: an analogue of interval graphs, J. Graph Theory 13 (1989) 189–202. [19] B.L. Shader, Biclique Partitions and Tournament Matrices, Ph.D. Thesis, University of Wisconsin-Madison, 1990. [20] V.L. Watts, Boolean rank of Kronecker products, Linear Algebra Appl. 336 (2001) 261–264. [21] D.B. West, Short proofs for interval digraphs, Discrete Math. 178 (1998) 287–292.
