On opposition graphs, coalition graphs, and bipartite permutation graphs

Discrete Applied Mathematics 168 (2014) 26–33

Contents lists available at ScienceDirect

Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam

On opposition graphs, coalition graphs, and bipartite permutation graphs Van Bang Le Universität Rostock, Institut für Informatik, Rostock, Germany

article

info

Article history: Received 24 September 2012 Received in revised form 16 November 2012 Accepted 23 November 2012 Available online 20 December 2012 Keywords: Perfectly orderable graph One-in-one-out graph Bipartite permutation graph Opposition graph Coalition graph

abstract A graph is an opposition graph, respectively, a coalition graph, if it admits an acyclic orientation which puts the two end-edges of every chordless 4-vertex path in opposition, respectively, in the same direction. Opposition and coalition graphs have been introduced and investigated in connection to perfect graphs. Recognizing and characterizing opposition and coalition graphs still remain long-standing open problems. The present paper gives characterizations for co-bipartite opposition graphs and co-bipartite coalition graphs, and for bipartite opposition graphs. Implicit in our argument is a linear time recognition algorithm for these graphs. As an interesting by-product, we find new submatrix characterizations for the well-studied bipartite permutation graphs. © 2012 Elsevier B.V. All rights reserved.

1. Introduction and preliminaries Chvátal [5] proposed to call a linear order < on the vertex set of an undirected graph G perfect if the greedy coloring algorithm applied to each induced subgraph H of G gives an optimal coloring of H: consider the vertices of H sequentially by following the order < and assign to each vertex v the smallest color not used on any neighbor u of v, u < v . A graph is perfectly orderable if it admits a perfect order. Chvátal proved that < is a perfect order if and only if there is no chordless path with four vertices a, b, c , d and three edges ab, bc , cd (written P4 abcd) with a < b and d < c. He also proved that perfectly orderable graphs are perfect.1 The class of perfectly orderable graphs properly contains many important, classical classes of perfect graphs such as chordal graphs and comparability graphs. Perfectly orderable graphs have been extensively studied in the literature; see Hoàng’s comprehensive survey [12] for more information. In [6], Chvátal pointed out a somewhat surprising connection between perfectly orderable graphs and a well-known theorem in mathematical programming. A matrix M is a submatrix of a matrix N if M can be obtained from N by deleting some columns and rows in N; N is M-free if there exist permutations of the rows and columns of N such that the permuted matrix (which we will again denote by N) does not contain M as a submatrix. A 0/1 matrix is totally balanced if it does not contain, as a submatrix, the edge-vertex incidence matrix of  a cycle  of length at least three. The following characterization of totally balanced 0/1 matrices is well known; write Γ =

1 1

1 0

.

Theorem 1 ([1,14,16]). A 0/1 matrix is totally balanced if and only if it is Γ -free. The bimatrix MG = (mij ) of a bipartite graph G = (X , Y , E ) is a 0/1 matrix whose rows represent the vertices in the color class X and whose columns represent the vertices in the color class Y in such a way that mij = 1 if and only if the vertex represented by the ith row is adjacent to the vertex represented by the jth column. It can be verified (cf. also the proof of E-mail address: [email protected]. 1 A graph is perfect if the chromatic number and the clique number are equal in every induced subgraph. 0166-218X/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.dam.2012.11.020

V.B. Le / Discrete Applied Mathematics 168 (2014) 26–33

27

Theorem 5) that the complement G of G is perfectly orderable if and only if MG is Γ -free, and that G is chordal bipartite (that is, G does not contain any chordless cycle with at least six vertices) if and only if MG is totally balanced. Thus, Theorem 1 can be reformulated as follows: Theorem 2 ([1,6,14,16]). For a bipartite graph G = (X , Y , E ), the following statements are equivalent: (i) G is perfectly orderable. (ii) G is chordal bipartite. (iii) MG is Γ -free. In particular, recognizing if a co-bipartite graph (the complement of a bipartite graph) is perfectly orderable reduces to recognizing if a graph is chordal bipartite, which can be done in quadratic time (cf. [22]). In general, recognizing perfectly orderable graphs is NP-complete [18] (see also [11]). Also, no characterization of perfectly orderable graphs by forbidden induced subgraphs is known. These facts have motivated researchers to study subclasses of perfectly orderable graphs; see, e.g., [8,12,13] and the literature given there. Observe that a linear order < corresponds to an acyclic orientation by directing the edge xy from x to y if and only if x < y. Thus, a graph is perfectly orderable if and only if it admits an acyclic orientation such that no chordless path P4 is oriented of type 0 depicted in Fig. 1; equivalently, every P4 is oriented of type 1, 2, or 3. type 0

type 1

type 2

type 3

Fig. 1. Four types of oriented P4 .

One of the natural subclass of perfectly orderable graphs for which the recognition complexity, as well as an induced subgraph characterization are still unknown is the following (cf. [12,13]). Definition 1. A graph is a coalition graph if it admits an acyclic orientation such that every induced P4 abcd has the end-edges ab and cd oriented in the ‘same way’, that is, every oriented P4 is of type 2 or 3. Equivalently, a graph is a coalition graph if it admits an order < on its vertex set such that every induced P4 abcd has a < b if and only if c < d. In [12], coalition graphs are called one-in-one-out graphs. Examples of coalition graphs include comparability graphs, hence all bipartite graphs. A related graph class has been introduced by Olariu in [19]: Definition 2. A graph is an opposition graph if it admits an acyclic orientation such that every induced P4 abcd has the endedges ab and cd oriented ‘in opposition’, that is, every oriented P4 is of type 0 or 1. Equivalently, a graph is an opposition graph if it admits an order < on its vertex set such that every P4 abcd has a < b if and only if d < c. Olariu [19] proved that opposition graphs are perfect. He also conjectures [20] that not all opposition graphs are perfectly orderable. Examples of opposition graphs include all split graphs. The recognition and characterization problems for opposition graphs are still open. A natural subclass of opposition graphs consists of those admitting an acyclic orientation in which every P4 is oriented as type 1 (equivalently, every P4 is oriented as type 0) has been characterized by forbidden induced subgraphs in [10,13], and has been recognized in O(n3.376 ) time in [8]; n is the vertex number of the input graph. The purpose of the present paper is to find characterizations for co-bipartite coalition graphs and co-bipartite opposition graphs, similar to those stated in Theorem 2, and for bipartite opposition graphs. In doing so, we will find new characterizations for the well-studied bipartite permutation graphs. Definition 3. A graph is a permutation graph if there is some pair π1 , π2 of permutations of the vertex set such that there is an edge xy if and only if x precedes y in one permutation in {π1 , π2 } and y precedes x in the other permutation. A bipartite permutation graph is a permutation graph that is also bipartite. Bipartite permutation graphs admit several characterizations and can be recognized in linear time [21]; see [22] for more information on permutation and bipartite permutation graphs. The following characterizations of bipartite permutation graphs follow first from the fact that permutation graphs are exactly the comparability graphs which are also co-comparability graphs, and second, from Gallai’s subgraph characterization of comparability graphs [9,17] that C2k , k ≥ 3, and B1 , B2 and B3 depicted in Fig. 2 are the only minimal bipartite graphs which are not co-comparability graphs. Theorem 3 (Folklore). Let G be a bipartite graph. Then G is a permutation graph if and only if G is a comparability graph if and only if G is a {B1 , B2 , B3 }-free chordal bipartite graph.

28

V.B. Le / Discrete Applied Mathematics 168 (2014) 26–33

B1

B2

B3

Fig. 2. Forbidden bipartite graphs for permutation graphs.

Where, given a set of graphs F , a graph G is F -free if it does not contain any graph in F as an induced subgraph. For a set of matricesM , a  M-free matrix defined.  is similarly  Write I =

0 1

1 0

and

=

0 1

1 1

. The following bimatrix characterization of bipartite permutation graphs is given

in [3], and can also be derived from results in [21]; cf. also [15]. Theorem 4 ([3,21]). A bipartite graph G = (X , Y , E ) is a permutation graph if and only if MG is {I , Γ ,

}-free.

In Section 2 we will characterize co-bipartite opposition graphs and co-bipartite opposition graphs. It turns out that they are exactly the complements of bipartite permutation graphs, and that bipartite permutation graphs can be characterized as submatrices in their bimatrix. by avoiding only Γ and In Section 3 we will characterize bipartite opposition graphs by (infinitely many) forbidden induced subgraphs, and show how every tree opposition graph can be built up from the one-vertex path and the two-vertex path by a sequence of two simple operations. We consider only finite, simple, and undirected graphs. For a graph G, the vertex set is denoted by V (G) and the edge set is denoted by E (G). For a vertex u of a graph G, the neighborhood of u in G is denoted by NG (u) or simply N (u) if the context is clear, and the degree of u is deg(u) = |N (u)|. For a set U of vertices of a graph G, the subgraph of G induced by U is denoted by G[U ]. If u is a vertex of a graph G, then G − u is G[V (G) \ {u}]. For ℓ ≥ 1, let Pℓ denote a chordless path with ℓ vertices and ℓ− 1 edges, and for ℓ ≥ 3, let Cℓ denote a chordless cycle with ℓ vertices and ℓ edges. We write Pℓ u1 u2 . . . uℓ and Cℓ u1 u2 . . . uℓ u1 , meaning the chordless path with vertices u1 , u2 , . . . , uℓ and edges ui ui+1 , 1 ≤ i < ℓ, respectively, the chordless cycle with vertices u1 , u2 , . . . , uℓ and edges ui ui+1 , 1 ≤ i < ℓ, and uℓ u1 ; the edges u1 u2 and uℓ−1 uℓ of the path Pℓ (ℓ ≥ 3) are the end-edges of the path. In this paper, all paths Pℓ and all cycles Cℓ will always be induced. An orientation of an undirected graph G is a directed graph D(G) obtained from G by replacing each edge xy of G by either x → y or y → x (but not both). Recall that a graph is a comparability graph if it admits an acyclic orientation of its edges such that whenever we have a → b and b → c we also have a → c. Equivalently, a graph is a comparability graph if it admits an acyclic orientation which puts the two edges of every P3 in opposition. It follows that comparability graphs are coalition graphs. Complements of bipartite/comparability graphs are co-bipartite graphs/co-comparability graphs. For more information on the graph classes appearing in this paper, and for basic graph notions and definitions not given here, see, e.g., [2]. 2. Co-bipartite coalition and opposition graphs This section gives characterizations for co-bipartite opposition graphs and co-bipartite coalition graphs. In doing so, we will find new characterizations of bipartite permutation graphs. The following fact can be verified by inspection. Fact 1. The co-bipartite graphs C2k , k ≥ 3, and G1 = B1 , G2 = B2 , G3 = B3 depicted in Fig. 3 are minimal non-coalition graphs, as well as minimal non-opposition graphs.

G1

G2

G3

Fig. 3. Forbidden co-bipartite graphs for coalition graphs and opposition graphs.

The main result of this section is the following theorem; we will prove it without using Theorem 4.

V.B. Le / Discrete Applied Mathematics 168 (2014) 26–33

29

Theorem 5. For a bipartite graph G = (X , Y , E ), the following statements are equivalent: (i) (ii) (iii) (iv)

G is a coalition graph. G is an opposition graph. G is a bipartite permutation graph. MG is {Γ , }-free.

Proof. Let G = (X , Y , E ) be a bipartite graph. (i) ⇒ (iii) This is because of Fact 1 and Theorem 3. (iii) ⇒ (i) This is because of Theorem 3 and the fact that comparability graphs are coalition graphs. (i) ⇒ (iv) Assume G is a coalition graph, and consider an order < on V (G) such that every P4 abcd of G has a < b if and only if c < d. Order the rows and columns of MG according to <. Then MG has no Γ as a submatrix: if there are rows r1 < r2 and columns c1 < c2 such that MG (r1 , c1 ) = MG (r1 , c2 ) = MG (r2 , c1 ) = 1 and MG (r2 , c2 ) = 0, then G[r1 , r2 , c1 , c2 ] is the P4 r2 c1 r1 c2 , hence G[r1 , r2 , c1 , c2 ] is the P4 r1 r2 c2 c1 with r1 < r2 but c1 < c2 , a contradiction. as a submatrix: if there are rows r1 < r2 and columns c1 < c2 such that MG (r1 , c1 ) = 0, and Similarly, MG has no MG (r1 , c2 ) = MG (r2 , c1 ) = MG (r2 , c2 ) = 1, then G[r1 , r2 , c1 , c2 ] is the P4 r2 r1 c1 c2 with c1 < c2 but r1 < r2 , a contradiction. Thus, MG is {Γ , }-free. (iv) ⇒ (i) Assume that we can order the vertices of G, say, X = {x1 , x2 , . . . , xp } and Y = {y1 , y2 , . . . , yq } such that MG is {Γ , }-free. Set u < v if u ∈ X and v ∈ Y , or there exist i < j such that u = xi , v = xj or u = yi , v = yj . Consider a chordless P4 abcd in G. Let, without loss of generality, a, b ∈ X and c , d ∈ Y . Then MG (a, c ) = MG (a, d) = MG (b, d) = 1 and MG (b, c ) = 0. Now, if a < b then c < d, otherwise the rows and columns a, b, c , d would form a Γ in MG , and if b < a then d < c, otherwise the rows and columns a, b, c , d would form a in MG . Thus, G is a coalition graph. (ii) ⇒ (iii) This is because of Fact 1 and Theorem 3. (iii) ⇒ (ii) Assume that G = (X , Y , E ) is a bipartite permutation graph, and consider a permutation diagram of G: each vertex v ∈ X ∪ Y corresponds to a line segment ℓ(v) drawn between the two parallel lines such that two vertices are adjacent if and only if the corresponding line segments cross. Since the lines ℓ(x), x ∈ X , are pairwise non-crossing, we can order the vertices in X = {x1 , x2 , . . . , xp } such that their line segments ℓ(x1 ), ℓ(x2 ), . . . , ℓ(xp ) are from left to right. Also, let Y = {y1 , y2 , . . . , yq } such that ℓ(y1 ), ℓ(y2 ), . . . , ℓ(yq ) are from left to right. Then it can be verified that, for every chordless P4 xi xj ys yt in G,

ℓ(xi ) is to the left of ℓ(xj ) if and only if ℓ(yt ) is to the right of ℓ(ys ). On V (G), define u < v if u ∈ X and v ∈ Y , or u, v ∈ X and ℓ(u) is to the left of ℓ(v), or u, v ∈ Y and ℓ(u) is to the right of ℓ(v). Now, consider a chordless P4 abcd in G, and let, without loss of generality, a, b ∈ X and c , d ∈ Y . Then, a < b if and only if ℓ(a) is to the left of ℓ(b) if and only if ℓ(d) is to the right of ℓ(c ) if and only if d < c . That is, G is an opposition graph. We note that if we define u < v if u ∈ X and v ∈ Y , or both u, v are in X or in Y and ℓ(u) is to the left of ℓ(v), then we will get an alternative proof for the implication (iii) ⇒ (i). The proof of Theorem 5 is complete.  Note that, by reversing the row order of the matrices, Theorem 5 implies that bipartite permutation graphs can be  1 0 , respectively, only L and , as submatrices in their bimatrix, where L = 1 1

characterized by avoiding only Γ and and

=



1 0



1 1

.

Corollary 1. A bipartite graph G is a permutation graph if and only if MG is {Γ ,

}-free if and only if MG is {L,

}-free.

Note that bipartite permutation graphs can be recognized in linear time, hence the proof of Theorem 5 implies the following: Theorem 6. There is a linear-time algorithm that, given a bipartite graph G, decides if G is an opposition/coalition graph. Moreover, if G is an opposition/coalition graph, the algorithm also constructs an opposition/coalition orientation for G. 3. Bipartite opposition graphs This section gives a characterization for bipartite opposition graphs by forbidden induced subgraphs. Recall that all comparability graphs, hence all bipartite graphs are coalition graphs. Note that, however, even not all trees are opposition graphs.

30

V.B. Le / Discrete Applied Mathematics 168 (2014) 26–33

1

2

1

2

3

4

T1

5

6

1

T3

1

2

3

4

T2

2k

Tk

Fig. 4. Forbidden trees for opposition graphs.

3.1. Tree opposition graphs For more clarity, we first characterize tree opposition graphs. Let k ≥ 1 be an integer. The tree Tk is obtained from two disjoint P4 s by connecting a vertex of degree 2 in one P4 and a vertex of degree 2 in the other P4 with a path on 2k vertices; see also Fig. 4. The following fact can be verified by inspection. Fact 2. The trees Tk , k ≥ 1, depicted in Fig. 4 are minimal non-opposition graphs. Two distinct vertices u and v of a graph G are twins if NG (u) = NG (v). If u, v are twins in G, then, clearly, G is an opposition graph if and only if G − u is an opposition graph. We will often make use of this fact in the reminder of this section. In a tree T , a big vertex is a vertex of degree at least 3 having at least two non-leaf neighbors. A leaf v of T is an odd leaf (an even leaf ) if the distance in T between v and any big vertex in T is odd (respectively, even). In a tree without big vertices, any leaf is both an odd leaf and an even leaf. For convenience, we also consider the vertex of the trivial tree, the one-vertex path P1 , as a leaf (which is then both an odd and an even leaf). Theorem 7. For a tree, the following statements are equivalent: (i) (ii) (iii) (iv)

T is an opposition graph. T is Tk -free for every k ≥ 1. The distance between any two big vertices in T is even. Every leaf in T is an odd leaf or an even leaf.

Proof. The equivalence (ii) ⇔ (iii) ⇔ (iv) are easy to see. The implication (i) ⇒ (ii) follows directly from Fact 2. It remains to prove (ii) ⇒ (i). Let T be a tree without induced Tk , k ≥ 1. We may assume that T has no twins. If T is a path, T is obvious an opposition graph. So, let u be some vertex of degree at least 3. Let Ni (u) be the set of all vertices at distance exactly i to u. Notice that N0 (u) = {u}, |N1 (u)| = deg(u) ≥ 3, and, as T has no twins, |N2 (u)| ≥ 2. Claim: If i is odd and v ∈ Ni (u), then deg(v) ≤ 2. Proof of the Claim: Assume that, for some odd i and some v ∈ Ni (u), deg(v) ≥ 3. Then v has two distinct neighbors x, y in Ni+1 (u). As x, y are not twins in G, x or y has a neighbor in Ni+2 (u). By symmetry, let x have a neighbor z ∈ Ni+2 (u). Let P be the u, v -path in T , and consider two distinct vertices w1 , w2 in N1 (u) \ V (P ). As w1 , w2 are not twins in T , w1 has a neighbor w3 ∈ N2 (u) \ (V (P ) ∪ {x, y}), say. But then w1 , w2 , w3 , x, y, z together with P induce a Tk with k = i+21 . The Claim follows. Consider now the following (acyclic) orientation D(T ) of T . Let xy be an edge with x ∈ Ni (u) and y ∈ Ni+1 (u).

• If i = 0 mod 4 or i = 1 mod 4, orient the edge xy ‘forward’, i.e., x → y is the corresponding directed edge in D(T ). • Otherwise, orient the edge xy ‘backward’, i.e., y → x is the corresponding directed edge in D(T ). Then, clearly, in D(T ) • for any even i, any vertex in Ni (u) is a sink or a source, and • for any odd i, any vertex in Ni (u) with degT (u) = 2 is a mixed vertex. This implies immediately that the end-edges of every induced P4 with vertices in four levels Ni (u), Ni+1 (u), Ni+2 (u), Ni+3 (u) are directed in opposition. Let P abcd be an induced P4 in T with a ∈ Ni+1 (u), b ∈ Ni (u), c ∈ Ni+1 (u), d ∈ Ni+2 (u). In this case, by the Claim, i must be even, and the properties above again imply that the end-edges ab and cd of P are directed in opposition. We have seen that D(T ) is an opposition orientation of T , and the proof of Theorem 7 is complete.  Implicit in the proof of Theorem 7 is a linear-time recognition algorithm that decides if a given tree is an opposition graph or finds a forbidden induced tree Tk . We now are going to describe how every tree opposition graph can be built up from the one-vertex path P1 and the twovertex path P2 by a sequence of two operations defined in Definition 4. A duplication of a vertex v in a graph G is the graph obtained from G by adding a new vertex v ′ and edges v ′ w, w ∈ NG (v). Thus, v and v ′ are twins in the new graph. Theorem 7 and its proof led to the following notion.

V.B. Le / Discrete Applied Mathematics 168 (2014) 26–33

31

Definition 4. The class of trees T is defined recursively as follows. (t1) P1 ∈ T , P2 ∈ T . (t2) If T ∈ T is a non-trivial tree, then the tree obtained from T by duplicating any leaf of T is in T . (t3) If T (v1 ) ∈ T , . . . , T (vd ) ∈ T , d ≥ 2, such that each vi is an odd leaf in T (vi ), then the tree obtained from T (v1 ), . . . , T (vd ) by adding a new vertex v and edges vvi , 1 ≤ i ≤ d, is in T . Theorem 8. A tree T is an opposition graph if and only if T ∈ T . Proof. First, let T be a tree opposition graph. We will show by induction that T ∈ T . By (t1) in Definition 4, we may assume that T has more than two vertices. We consider three cases.

• T is a path. Let v be a non-leaf vertex of T with neighbors v1 , v2 , and let T (v1 ) and T (v2 ) be the trees in T − v containing v1 , respectively, v2 . By induction, T (v1 ), T (v2 ) ∈ T , hence, by (t3), T ∈ T . • T has twins v, v ′ . Then, by induction, T ′ = T − v ′ ∈ T , hence, by (t2), T ∈ T . • T is not a path and have no twins. Let v be a vertex of degree d ≥ 3, and let v1 , . . . , vd be the neighbors of v in T . Let T (vi ) be the tree in T − v containing vi , 1 ≤ i ≤ d. By induction, T (vi ) ∈ T . As T has no twins, at most one of the trees T (vi ) is an one-vertex tree. Hence, v is a big vertex in T . Moreover, every vi is a leaf in T (vi ) (otherwise, if vi is non-leaf in T (vi ), then, as T has no twins, vi is a big vertex in T . But the big vertices vi and v have distance one in T , contradicting Theorem 7). Furthermore, every vi is an odd leaf of T (vi ) (otherwise, if v ′ ∈ T (vi ) is a big vertex at even distance to vi , then the two big vertices v and v ′ have odd distance in T , a contradiction again). Thus, by (t3), T ∈ T . Now, let T ∈ T . We show, by induction again, that T is an opposition graph. Thus, we may assume that T has more than two vertices. According to (t2) and (t3) in Definition 4 we have two cases.

• There exists a tree T ′ ∈ T with leaf v ′ and T is obtained from T ′ by duplicating v ′ . Then, by induction, T ′ is an opposition graph, hence clearly, T is an opposition graph.

• There are trees T (v1 ) ∈ T , . . . , T (vd ) ∈ T , d ≥ 2, such that each vi is an odd leaf in T (vi ), and T is obtained from T (v1 ), . . . , T (vd ) by adding a new vertex v and edges vv1 , . . . , vvd . Then, by induction, every T (vi ) is an opposition graph. Since every vi is an odd leaf in T (vi ), it follows easily from Theorem 7 that T is an opposition graph. The proof of Theorem 8 is complete.



3.2. Bipartite opposition graphs We now are going to characterize bipartite opposition graphs. It is interesting to remark that it was conjectured by Chvátal [7] and is implied by the strong perfect graph theorem,2 that if the term ‘acyclic’ in Definition 2 of opposition graphs is dropped, the larger class is still a class of perfect graphs. Chvátal [7] proposed to call these graphs generalized opposition graphs: Definition 5. A graph is a generalized opposition graph if it admits an orientation such that every oriented P4 is of type 0 or 1. The co-bipartite graphs C2k , k ≥ 3, are examples of generalized opposition graphs that are non-opposition graphs. Clearly, a tree is an opposition graph if and only if it is a generalized opposition graph. As we will see, this fact can be extended for bipartite graphs: bipartite opposition graphs and bipartite generalized opposition graphs coincide. The following fact can be verified by inspection. Fact 3. The chordless even cycles C4k+2 and the bipartite graphs A and B depicted in Fig. 5 are not generalized opposition graphs.

A

B

Fig. 5. Forbidden bipartite graphs for (generalized) opposition graphs.

Theorem 9. For a bipartite graph G, the following statements are equivalent: (i) G is an opposition graph. (ii) G is a generalized opposition graph. (iii) G is C4k+2 -free, Tk -free, k ≥ 1, and {A, B}-free; see Figs. 4 and 5.

2 The strong perfect graph theorem [4] states that graphs without chordless cycles of odd length at least five and without complements of such a cycle are perfect.

32

V.B. Le / Discrete Applied Mathematics 168 (2014) 26–33

Proof. The implication (i) ⇒ (ii) is obvious, and (ii) ⇒ (iii) follows directly from Facts 2 and 3. We will prove (iii) ⇒ (i) by induction. Let G be an {A, B}-free bipartite graph without induced C4k+2 , Tk , k ≥ 1. We may assume that G is connected and has no twins. A domino is obtained from the cycle C6 by adding a chord between two distance 3-vertices. Claim 1. If G contains an induced domino D, then G = D. Proof of Claim 1. Let D be an induced domino in G with vertices d1 , . . . , d6 and edges d1 d2 , d2 d3 , d3 d4 , d4 d5 , d5 d6 , d6 d1 and d2 d5 . Note that, as G is bipartite, the neighborhood in D of any vertex of G is a subset of {d1 , d3 , d5 } or else a subset of {d2 , d4 , d6 }. First, assume that there is some vertex u ∈ V (G) \ V (D) with N (u) ∩ V (D) = {d1 , d3 , d5 }. As d2 and u are no twins, there exists some vertex v with v d2 ∈ E (G) and v u ̸∈ E (G), say. Since G[d1 , d2 , d4 , d5 , u, v] ̸= A, v d4 ∈ E (G). Then, since G[d3 , d4 , d5 , d6 , u, v] ̸= A, v d6 ∈ E (G). But now d1 , u, d3 , d4 , v and d6 induce a C6 in G. Thus, no vertex apart from d2 is adjacent to all d1 , d3 , d5 , and by symmetry, no vertex apart from d5 is adjacent to all d2 , d4 , d6 . Next, assume that there is some vertex u ∈ V (G) \ V (D) with d5 ∈ N (u) ∩ V (D). As G is A-free, ud1 ∈ E (G) or ud3 ∈ E (G). By symmetry, N (u) ∩ D = {d1 , d5 }, say. As d6 and u are no twins, there exists some vertex v with v d6 ∈ E (G) and v u ̸∈ E (G), say. Since G[d1 , d4 , d5 , d6 , u, v] ̸= A, v d4 ∈ E (G). Hence v d2 ̸∈ E (G), and v, d4 , d3 , d2 , d1 , d6 induce a C6 in G. Thus, no vertex outside D is adjacent to d5 , and by symmetry, no vertex outside D is adjacent to d2 . Now, if G ̸= D, there must be a vertex u ∈ V (G) \ V (D) adjacent to a vertex di , i ̸= 2, 5. By symmetry, ud1 ∈ E (G), say. But then d1 , d2 , d3 , d5 , d6 , u induce an A (if ud3 ̸∈ E (G)) or d1 , u, d3 , d4 , d5 , d6 induce a C6 (otherwise). Claim 1 follows.  Claim 2. If G contains an induced C4 , then G contains an induced domino. Proof of Claim 2. Otherwise, two nonadjacent vertices of an induced C4 in G would be twins or else G would contain an induced A.  By Claims 1 and 2, G is a domino and it is obvious that G is an opposition graph, or else G has no C4 . So, we may assume that G is C4 -free. If G is a tree, then G is an opposition graph by Theorem 7. Hence we may assume that G contains a cycle, and let C v1 v2 . . . v4k v1 be an induced cycle with smallest k ≥ 2. Claim 3. For every vertex u ∈ V (G) \ V (C ), |N (u) ∩ V (C )| ≤ 1. Proof of Claim 3. Assume, by contradiction, |N (u) ∩ V (C )| ≥ 2 for some u ∈ V (G) \ V (C ). For simplicity, say v1 ∈ N (u) ∩ V (C ). Let vj be another neighbor of u on C such that v1 , . . . , vj together with u induce a cycle Cj+1 . As G is C4 -free, and by the choice of k, 4k ≤ j + 1 < 4k, a contradiction. Hence Claim 3.  Note that, as for every ℓ, C4ℓ is obviously an opposition graph, we may assume that G ̸= C . Thus, as G is connected, we may assume that there exists a vertex v ∈ V (G) \ V (C ) adjacent to v1 , say. By Claim 3, N (v) ∩ V (C ) = {v1 }. Claim 4. For every even i, deg(vi ) = 2. Proof of Claim 4. Assume, by contradiction, that for some even index i, vi is adjacent to a vertex w outside C . By symmetry, we may assume that i ≤ 2k. Then wv ̸∈ E (G), otherwise vv1 . . . vi wv would be a cycle of length i + 2 < 4k, contradicting the choice of C . Now, by Claim 3, if i + 2 < 4k − 2, then v4k−1 , v4k , v1 , . . . , vi , vi+1 vi+2 together with v and w induce a Ti/2 , a contradiction. If i + 2 ≥ 4k − 2, then k = 2 and i = 4, and C together with v and w induce a B, a contradiction again. Hence Claim 4.  Now, by induction, G′ = G −v4 has an opposition orientation G′ = D(G′ ). Without loss of generality, say v1 → v2 ∈ E (G′ ). Note that this together with Claim 3 imply that

v3 and v7 are sinks, and v5 is a source in D′ . Let D(G) be the orientation obtained from D′ by giving v3 v4 the orientation v4 → v3 and v4 v5 the orientation v5 → v4 . It is clear, by the fact above and by Claim 4, D(G) is an opposition orientation of G. Hence (i), and the proof of Theorem 9 is complete. 

V.B. Le / Discrete Applied Mathematics 168 (2014) 26–33

33

Note that twins can be detected in linear time, hence the proofs of Theorems 7 and 9 imply the following: Theorem 10. There is a linear-time algorithm that, given a bipartite graph G, decides if G is a (generalized) opposition graph or finds a forbidden induced subgraph in G. Moreover, if G is an opposition graph, the algorithm also constructs an opposition orientation for G. 4. Concluding remarks Recognizing and characterizing coalition graphs and opposition graphs are long-standing open problems. While coalition graphs form a natural subclass of perfectly orderable graphs, it is not known if there exists an opposition graph that is not perfectly orderable. In this paper we have shown that co-bipartite coalition graphs and co-bipartite opposition graphs coincide, and they are exactly the complements of bipartite permutation graphs. Thus, co-bipartite coalition graphs and co-bipartite opposition graphs can be recognized in linear time, and co-bipartite opposition graphs are perfectly orderable. We also characterized bipartite opposition graphs by means of infinitely many forbidden induced subgraphs (recall that all bipartite graphs are coalition graphs). A possible next step towards recognizing and characterizing coalition graphs and opposition graphs is to extend the results of this paper to larger graph classes properly containing all co-bipartite graphs, respectively, all bipartite graphs (in case of opposition graphs). Acknowledgments The author thanks the unknown referees for their many helpful comments and suggestions that led to improvements in the presentation of the paper. Especially, the author is grateful to one of the referees who pointed out a gap in an earlier version of Theorem 9. References [1] Richard P. Anstee, Martin Farber, Characterizations of totally balanced matrices, Journal of Algorithms 5 (1984) 215–230. [2] Andreas Brandstädt, Van Bang Le, Jeremy P. Spinrad, Graph Classes: A Survey, in: Monographs on Discrete Math. Appl., vol. 3, SIAM, Philadelphia, 1999. [3] Lin Chen, Yaacov Yesha, Efficient parallel algorithms for bipartite permutation graphs, Networks 23 (1993) 29–39. [4] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas, The strong perfect graph theorem, Annals of Mathematics 164 (2006) 51–229. [5] Vašek Chvátal, Perfectly ordered graphs, in: C. Berge, V. Chvátal (Eds.), Topics on Perfect Graphs, in: Annals of Discrete Mathematics, vol. 21, 1984, pp. 63–65. [6] Vašek Chvátal, Which claw-free graphs are perfectly orderable? Discrete Applied Mathematics 44 (1993) 39–63. [7] Vašek Chvátal, Generalized opposition graphs, Perfection of Special Classes of Berge Graphs, http://www.cs.rutgers.edu/∼chvatal/perfect/problems. html, August 24, 2000. [8] Elaine M. Eschen, Julie L. Johnson, Jeremy P. Spinrad, R. Sritharan, Recognition of some perfectly orderable graph classes, Discrete Applied Mathematics 128 (2003) 355–373. [9] Tibor Gallai, Transitiv orientierbare graphen, Acta Mathematica Academiae Scientiarum Hungarica 18 (1967) 25–66. [10] Alain Hertz, Bipolarizable graphs, Discrete Mathematics 81 (1990) 25–32. [11] Chính T. Hoàng, On the complexity of recognizing a class of perfectly orderable graphs, Discrete Applied Mathematics 66 (1996) 219–226. [12] Chính T. Hoàng, Perfectly orderable graphs: a survey, in: J.L. Ramírez Alfonsín, B.A. Reed (Eds.), Perfect Graphs, Wiley Interscience, New York, 2001. [13] Chính T. Hoàng, Bruce A. Reed, Some classes of perfectly orderable graphs, Journal of Graph Theory 13 (1989) 445–463. [14] Alan J. Hoffman, A.W.J. Kolen, M. Sakarovitch, Totally-balanced and greedy matrices, SIAM Journal on Algebraic and Discrete Methods 6 (1985) 721–730. [15] Bettina Klinz, Rüdiger Rudolf, Gerhard J. Woeginger, Permuting matrices to avoid forbidden submatries, Discrete Applied Mathematics 60 (1995) 223–248. [16] Anna Lubiw, Doubly lexical orderings of matrices, SIAM Journal on Computing 16 (1987) 854–879. [17] Frédéric Maffray, Myriam Preissmann, A translation of Tibor Gallai’s article ‘Transitiv orientierbare Graphen’, in: J.L. Ramírez Alfonsín, B.A. Reed (Eds.), Perfect Graphs, Wiley Interscience, New York, 2001. [18] Matthias Middendorf, Frank Pfeiffer, On the complexity of recognizing perfectly orderable graphs, Discrete Mathematics 80 (1990) 327–333. [19] Stephan Olariu, All variations on perfectly orderable graphs, Journal of Combinatorial Theory. Series B 45 (1988) 150–159. [20] Stephan Olariu, Berlin, Rostock, September 1997 (Private communication). [21] Jeremy P. Spinrad, Andreas Brandstädt, Lorna Stewart, Bipartite permutation graphs, Discrete Applied Mathematics 18 (1987) 279–292. [22] Jeremy P. Spinrad, Efficient Graph Representations, in: Fields Institute Monographs, vol. 19, American Mathematical Society, Providence, 2003.