Representation characterizations of chordal bipartite graphs

Journal of Combinatorial Theory, Series B 96 (2006) 673–683

www.elsevier.com/locate/jctb

Representation characterizationsof chordal bipartite graphs

Jing Huang

Department of Mathematics and Statistics, University of Victoria, PO Box 3045, Victoria, BC, Canada V8W 3P4

Received 24 December 2003

Available online 8 February 2006

Abstract

A bipartite graph is chordal bipartite if it does not contain an induced cycle of length at least six. We givethree representation characterizations of chordal bipartite graphs. More precisely, we show that a bipartitegraph is chordal bipartite if and only if the complement is the intersection graph of a family of pairwisecompatible claws in a weighted hypercircle. (A hypercircle is a graph which consists of internally vertexdisjoint paths between two distinguished vertices, and a claw in a hypercircle is a connected subgraph con-taining exactly one of the two distinguished vertices.) We also introduce two classes of bipartite graphs,both containing interval bigraphs and interval containment bigraphs. They are compatible subtree intersec-tion bigraphs and compatible subtree containment bigraphs. We show that these two classes are identical tothe class of chordal bipartite graphs.© 2006 Elsevier Inc. All rights reserved.

Keywords: Chordal bipartite graph; Strongly chordal graph; Ferrers dimension; Interval bigraph; Circular arc graph;Characterization

1. Introduction

A graph G is an intersection graph if there is a family of sets Sv, v ∈ V (G), such that twovertices u, v are adjacent if and only if Su, Sv intersect. If each set in the family can be chosento be an interval on the real line, then G is called an interval graph [14]; if each can be chosento be a circular arc on a fixed circle, then G is called a circular arc graph [29].

A graph is chordal if it does not contain an induced cycle of length at least four. Chordalgraphs were introduced by Hajnal and Surányi [13] and enjoy many nice structural properties.

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674 J. Huang / Journal of Combinatorial Theory, Series B 96 (2006) 673–683

For instance, every induced subgraph of a chordal graph contains a simplicial vertex, that is,a vertex whose neighbourhood induces a complete subgraph, cf. [4,9,22]. Chordal graphs are theintersection graphs of subtrees in trees, cf. [2,10,30].

Interval graphs are all chordal and, according to Lekkerkerker and Boland [19], they are pre-cisely those chordal graphs which do not contain asteroidal triples. An asteroidal triple in agraph is a set of three vertices such that any two of them are joined by a path which contains noneighbours of the third vertex.

Another class of intersection graphs, called strongly chordal graphs, were introduced by Far-ber [5,6]. These are the graphs in which every induced subgraph contains a simple vertex, thatis, a vertex whose neighbours can be linearly ordered by inclusion of their closed neighbour-hoods. Since every simple vertex is simplicial, all strongly chordal graphs are chordal. In [5],it is shown that strongly chordal graphs are the intersections graphs of compatible subtrees inrooted weighted trees (the definitions are given in Section 3).

Let H be a bipartite graph with a fixed bipartition (X,Y ). Then H is called an intersectionbigraph if there exists a family of sets Sv , v ∈ X ∪ Y , such that for x ∈ X and y ∈ Y , x, y areadjacent if and only if Sx,Sy intersect. As above, if all the sets can be chosen to be intervals, thenH is called an interval bigraph; if all can be chosen to be circular arcs on a fixed circle, then H

is called a circular arc bigraph.Interval bigraphs were first studied in [16]. An equivalent concept, in terms of digraphs, was

introduced in [23]: A digraph D is an interval digraph if there exists a family of ordered pair ofintervals (Iv, Jv), v ∈ V (D), such that uv is an arc of D if and only if Iu, Jv intersect. Let D∗ bethe associated bipartite graph obtained from D by replacing each vertex v of D by two verticesv′, v′′, and each arc uv of D by the edge u′v′′. Then it is clear from the definitions that D is aninterval digraph if and only if D∗ is an interval bigraph. Similarly, if H is a bipartite graph withbipartition (X,Y ), we can orient all edges from X to Y , and observe that the resulting digraph isan interval digraph if and only if H is an interval bigraph, cf. [20]. Sen et al. [24] (cf. also [8])proved that a graph G is an interval graph if and only if the reflexive symmetric digraph (i.e.,the digraph obtained from G by replacing each edge with two opposite arcs and adding a loop ateach vertex) is an interval digraph.

A bipartite graph is called chordal bipartite if it does not contain an induced cycle of length atleast six. Chordal bipartite graphs were introduced by Golumbic and Goss [11], as a natural bi-partite analogue of chordal graphs. Every non-trivial chordal bipartite graph contains a simplicialedge, that is, a pair of adjacent vertices whose neighbours induce a complete bipartite subgraph.In fact, a bipartite graph is chordal bipartite if and only if every non-trivial induced subgraphcontains a simplicial edge, cf. [11].

In spite of many rich structural properties of chordal bipartite graphs, it seems challenging tofind an intersection graph characterization of this class of graphs, cf. [27]. In this paper, we provethat a bipartite graph H is chordal bipartite if and only if the complement of H is the intersectiongraph of a family of compatible claws in a weighted hypercircle. A hypercircle is a graph whichconsists of internally vertex disjoint paths between two distinguished vertices, and a claw in ahypercircle is a connected subgraph containing exactly one of the two distinguished vertices.We also introduce two classes of bipartite graphs—compatible subtree intersection bigraphs andcompatible subtree containment bigraphs. We show that these two classes are identical to theclass of chordal bipartite graphs.

Bipartite graphs whose complements are circular arc graphs have been studied in [7,17,26,28]. In [7], it is shown that they are precisely those chordal bipartite graphs which do notcontain edge-asteroidal triples. An edge-asteroid is a set of edges e0, e1, . . . , e2k such that, for

J. Huang / Journal of Combinatorial Theory, Series B 96 (2006) 673–683 675

each i = 0,1, . . . ,2k, there is a path joining ei and ei+1, and containing both ei and ei+1, thatavoids the neighbours of ei+k+1. Since interval bigraphs are chordal bipartite and cannot containedge-asteroids, their complements are circular arc graphs. In fact, a bipartite graph is an intervalbigraph if and only if its complement is a circular arc graph, which can be represented by a fam-ily of circular arcs, in which no two arcs together cover the whole circle [18]. We shall give inthis paper a short proof of this statement.

Several variations of interval bigraphs and circular arc bigraphs have been studied in [24].A bipartite graph H with a fixed bipartition (X,Y ) is called an interval containment bigraph(respectively circular arc containment bigraph) with respect to (X,Y ) if there is a family ofintervals (respectively circular arcs) Iv , v ∈ X ∪ Y , such that for x ∈ X and y ∈ Y , xy ∈ E(H) ifand only if Ix contains Iy . We make two remarks on the definitions. First, the partite sets X,Y

are not symmetric in the definitions. There are circular arc containment bigraphs with respect to(X,Y ), which are not circular arc containment bigraphs with respect to (Y,X). However, we shallshow that if H is an interval containment bigraph with respect to (X,Y ), then H is an intervalcontainment bigraph with respect to (Y,X), that is, there is a family of intervals Jv, v ∈ X ∪ Y ,such that for x ∈ X and y ∈ Y , xy ∈ E(H) if and only if Jx is contained in Jy . Second, asobserved in [24], it is always possible to choose intervals (respectively circular arcs) so that theendpoints are all distinct and so in particular the containments are all proper containments.

Interval bigraphs and interval containment bigraphs are related to Ferrers bigraphs. A bi-partite graph H with a bipartition (X,Y ) is called a Ferrers bigraph if the vertices of X areneighbourhood-comparable, that is, for any x, x′ ∈ X either N(x) ⊆ N(x′) or N(x′) ⊆ N(x).This is equivalent to say that the vertices of Y are neighbourhood-comparable. Ferrers bigraphsare the bipartite analogues of Ferrers digraphs, introduced independently by Guttman [12] andRiguet [21]. Every bipartite graph is an intersection of a finitely many Ferrers bigraphs. Indeed,suppose that H is an arbitrary bipartite graph with a bipartition (X,Y ). For each non-adjacentpair x ∈ X,y ∈ Y , let Hx,y be the bipartite graph obtained from H by adding all missing edgesbetween X and Y except the one between x and y. It is easy to see that each Hx,y is a Ferrersbigraph and the intersection of them is precisely H . The Ferrers dimension of a bipartite graphH is the minimum number of Ferrers bigraphs whose intersection is H .

Sen et al. [24] proved that a bipartite graph is an interval containment bigraph if and only ifit has Ferrers dimension at most two. Bipartite graphs of Ferrers dimension at most two havealso been characterized by Cogis [3] in terms of obstructions in their adjacency matrices. Thischaracterization is observed by Sen et al. [25] to be equivalent to a characterization of bipartitegraphs whose complements are circular arc graphs, given in [17]. Therefore, a bipartite graph isan interval containment bigraph if and only if its complement is a circular arc graph. Combin-ing this with the characterization of interval bigraphs [18] mentioned above, we see that everyinterval bigraph is an interval containment bigraph.

In the next section, we shall give short proofs of theorems which describe relations amonginterval bigraphs, interval containment bigraphs, bipartite graphs of Ferrers dimension at mosttwo, and bipartite graphs whose complements are circular arc graphs. In Section 3, we shallextend the characterizations of interval containment bigraphs to representation characterizationsof chordal bipartite graphs.

2. Characterizations of interval bigraphs and interval containment bigraphs

We begin with the following lemma which has appeared in different forms in [3,24]. Weinclude a short proof for the sake of completeness.


Lemma 2.1. Let H be a bipartite graph with bipartition (X,Y ). Then H is a Ferrers bigraph ifand only if there exists a function f :X ∪ Y → R such that for x ∈ X and y ∈ Y , xy ∈ E(H) ifand only if f (x) � f (y).

Proof. Suppose that there exists a function f :X ∪ Y → R such that for x ∈ X and y ∈ Y , xy ∈E(H) if and only if f (x) � f (y). Then for any x, x′ ∈ X, N(x) ⊇ N(x′) when f (x) � f (x′).Hence H is a Ferrers bigraph.

For the converse suppose that H is a Ferrers bigraph, that is, the vertices of X (respectively Y )are neighbourhood-comparable. Define f :X ∪ Y → R as follows: For each x ∈ X, f (x) =min{d(v): v ∈ N(x)} (when N(x) = ∅, f (x) = |X|), and for each y ∈ Y , f (y) = d(y). Supposethat x ∈ X and y ∈ Y . If xy ∈ E(H), then the definition of f implies f (x) � f (y). Suppose thatxy /∈ E(H). If N(x) = ∅, then f (x) = |X| > d(y) = f (y). So assume that N(x) = ∅ and letv ∈ N(x) be a vertex such that f (x) = d(v). Since v and y are neighbourhood-comparable andx ∈ N(v) − N(y), we must have N(v) ⊃ N(y). Therefore f (x) = d(v) > d(y) = f (y). �

Lemma 2.1 can be easily extended as follows.

Theorem 2.2. A bipartite graph H with bipartition (X,Y ) is of Ferrers dimension at most k ifand only if there exist k functions fi :X ∪Y → R, i = 1,2, . . . , k, such that for x ∈ X and y ∈ Y ,xy ∈ E(H) if and only if fi(x) � fi(y) for each i = 1,2, . . . , k.

The characterizations of interval containment bigraphs given in the following theorem may besummarized from results in [17,23–25]. Here we provide yet another short proof of the theorem.

Theorem 2.3. Let H be a bipartite graph with a fixed bipartition (X,Y ). Then the followingstatements are equivalent:

(i) H is an interval containment bigraph with respect to (X,Y );(ii) there is a family of intervals Jv , v ∈ X ∪ Y , such that

• all intervals Jv intersect;• for any x ∈ X and y ∈ Y , x and y are adjacent if and only if Jx contains Jy ;

(iii) H is of Ferrers dimension at most two;(iv) the complement of H is a circular arc graph.

Proof. (i) ⇒ (ii) Since H is an interval containment bigraph with respect to (X,Y ), there is afamily of intervals Iv = [av, bv], v ∈ X ∪ Y , such that for x ∈ X and y ∈ Y , x, y are adjacentif and only if ax < ay and by < bx . Assume without loss of generality that av, bv > 0 for allv ∈ X ∪ Y . Let Jv = [−1/av, bv] for all v ∈ X ∪ Y . Clearly, all intervals Jv intersect as they allcontain the point zero. For all x ∈ X and y ∈ Y , ax < ay if and only if −1/ax < −1/ay . Hencefor all x ∈ X and y ∈ Y , x, y are adjacent if and only if Jx contains Jy .

(ii) ⇒ (iii) Suppose that there exists a family of intervals Jv = [av, bv], v ∈ X ∪ Y , satisfyingthe two properties itemized in (ii). Assume without loss of generality that av < 0 < bv for eachv ∈ X ∪ Y . Define f1(v) = av and f2(v) = 1/bv for each v ∈ X ∪ Y . Then for x ∈ X and y ∈ Y ,xy ∈ E(H) if and only if f1(x) = ax � ay = f1(y) and f2(x) = 1/bx � 1/by = f2(y). Henceby Theorem 2.2, H is of Ferrers dimension at most two.

(iii) ⇒ (iv) Suppose that H is of dimension at most two. By Theorem 2.2 there are twofunctions fi :X ∪ Y → R, i = 1,2, such that xy ∈ E(H) if and only if f1(x) � f1(y) and


f2(x) � f2(y). Let C be a circle with two specified points p,q diametrically opposed on C, andlet A (respectively B) be the open segment extending from p to q clockwise (counterclockwise).Assume the points of A (respectively B) are labeled by real numbers, i.e., A = R = B . Obtaina family of (open) circular arcs Fv, v ∈ X ∪ Y , as follows: For each x ∈ X, Fx extends fromthe point f2(x) ∈ B to the point f1(x) ∈ A in the clockwise direction, and for each y ∈ Y , Fy

extends from the point f1(y) ∈ A to the point f2(y) ∈ B . Thus Fx contains p for each x ∈ X andFy contains q for each y ∈ Y . For x ∈ X and y ∈ Y , Fx,Fy intersect if and only if f1(x) > f1(y)

or f2(x) > f2(y), i.e., x, y are not adjacent in H . This shows that the complement of H is acircular arc graph.

(iv) ⇒ (i) Suppose that the complement of H is a circular arc graph. Then it is easy to show(see, for example, [26]) that there exists a family of circular arcs Fv , v ∈ X ∪ Y , in a circle C

with two specified points p,q , such that

• each Fx with x ∈ X contains p but not q and each Fy with y ∈ Y contains q but not p;• x ∈ X and y ∈ Y are adjacent if and only if Fx and Fy are disjoint.

Let F ′x be the complement of Fx on C. Then F ′

x , x ∈ X, together with Fy , y ∈ Y , form a family ofintervals satisfying the property that x ∈ X and y ∈ Y are adjacent if and only if F ′

x contains Fy .Therefore H is an interval containment bigraph. �

Statement (iv) in Theorem 2.3 is symmetric with respect to the bipartition (X,Y ), so we have:

Corollary 2.4. Let H be a bipartite graph with a fixed bipartition (X,Y ). Then H is an intervalcontainment bigraph with respect to (X,Y ) if and only if H is an interval containment bigraphwith respect to (Y,X).

The next theorem characterizes interval bigraphs in a fashion similar to the above characteri-zations of interval containment bigraphs. The equivalence between (i) and (iv) has been provedin [18]. Our proof here is shorter.

Theorem 2.5. Let H be a bipartite graph with a fixed bipartition (X,Y ). Then the followingstatements are equivalent:

(i) H is an interval bigraph;(ii) there is a family of intervals Jv, v ∈ X ∪ Y , such that

• all intervals Jv intersect;• for any x ∈ X and y ∈ Y , Jx is not contained in Jy ;• for any x ∈ X and y ∈ Y , x, y are adjacent if and only if Jx contains Jy ;

(iii) there exist two functions fi :X ∪ Y → R, i = 1,2, such that for x ∈ X and y ∈ Y

• f1(x) � f1(y) or f2(x) � f2(y);• xy ∈ E(H) if and only if f1(x) � f1(y) and f2(x) � f2(y);

(iv) the complement of H is a circular arc graph representable by a family of circular arcs, notwo of which together cover the whole circle.

Proof. (i) ⇒ (ii) Since H is an interval bigraph, there is a family of intervals Iv = [av, bv],v ∈ X ∪ Y , such that for any x ∈ X and y ∈ Y , x and y are adjacent if and only if Ix and Iy

intersect, that is, ax � by and ay � bx . Without loss of generality, assume that av, bv > 0 and they


are all distinct for all v ∈ X ∪ Y . Let Jx = [−1/ax, bx] when x ∈ X, and let Jy = [−1/by, ay]when y ∈ Y . We show that the family of intervals satisfies the three properties itemized in (ii).Since all intervals contain the point zero, they all intersect. The third property follows from thefact that ax < by if and only if −1/ax < −1/by . To show the second property, suppose thatJx is contained in Jy for some x ∈ X and y ∈ Y . Then by < ax and bx < ay . Hence by < ay ,a contradiction.

(ii) ⇒ (iii) Suppose that there exists a family of intervals Jv = [av, bv], v ∈ X ∪ Y , satisfyingthe three properties itemized in (ii). The first property in (ii) allows to assume that av < 0 < bv foreach v ∈ X ∪ Y . Define f1(v) = av and f2(v) = 1/bv for each v ∈ X ∪ Y . The second propertyin (ii) ensures that either f1(x) � f1(y) or f2(x) � f2(y). By the third property in (ii), for x ∈ X

and y ∈ Y , xy ∈ E(H) if and only if ax � ay and 1/bx � 1/by , that is, f1(x) � f1(y) andf2(x) � f2(y).

(iii) ⇒ (iv) The proof of this is similar to that of (iii) ⇒ (iv) for Theorem 2.3. The firstproperty in (iii) ensures that the transformation will not result two arcs which together cover thewhole circle.

(iv) ⇒ (i) Suppose that (iv) is satisfied. Again according to [26], the complement of H can berepresented by a special family of circular arcs Fv , v ∈ X ∪ Y , on a circle C with two specifiedpoints p,q such that

• each Fx with x ∈ X contains p but not q , and each Fy with y ∈ Y contains q but not p;• for x ∈ Y and y ∈ Y , x, y are adjacent in H if and only if Fx , Fy are disjoint;• no two arcs together cover the whole circle.

Let (p, q) (respectively (q,p)) denote the segment extending from p to q (respectively from q

to p) clockwise. Assume that (p, q) = (0,1) and (q,p) = (−1,0). Thus each arc Fx withx ∈ X has its clockwise endpoint ax ∈ (p, q) = (0,1) and its counterclockwise endpoint bx ∈(q,p) = (−1,0), and each arc Fy with y ∈ Y has its clockwise endpoint cy ∈ (q,p) = (−1,0)

and its counterclockwise endpoint dy ∈ (p, q) = (0,1). We now define a family of inter-vals Iv , v ∈ X ∪ Y , as follows: For each x ∈ X let Ix = [bx,−1/ax] and for each y ∈ Y letIy = [cy,−1/dy]. For any x ∈ X and y ∈ Y , Fx and Fy are disjoint if and only if ax < dy andcy < bx , i.e., the two intervals Ix = [ax,−1/bx] and Iy = [−1/cy, dy] intersect. This shows thatH is an interval bigraph. �Corollary 2.6. Every interval bigraph is an interval containment bigraph.

For a graph G, let B(G) be the bipartite graph with vertex set {v′, v′′: v ∈ V (G)} and edgeset {v′v′′: v ∈ V (G)} ∪ {u′v′′, u′′v′: uv ∈ E(G)}. In the case when H = B(G) for some G, allstatements in Theorems 2.3 and 2.5 become equivalent.

Theorem 2.7. The following are equivalent for a graph G:

(i) G is an interval graph;(ii) B(G) is an interval bigraph;

(iii) B(G) is an interval containment bigraph;(iv) B(G) is of Ferrers dimension at most two;(v) the complement of B(G) is a circular arc graph;


(vi) the complement of B(G) is a circular arc graph representable by a family of circular arcs,no two of which together cover the whole circle.

Proof. The equivalence of (i) and (v) has been proved in [8]. The equivalence of (ii) and (vi)follows from Theorem 2.5. It is easy to see that (i) implies (ii) and that (vi) implies (v). Finally,the equivalence of (iii)–(v) follows from Theorem 2.3. �

We note that the equivalence of (i) and (ii) has also been proved in [24].As mentioned in Section 1, a forbidden substructure characterization has been given in [7] for

bipartite graphs whose complements are circular arc graphs and hence for interval containmentbigraphs. No such a characterization is known for interval bigraphs.

3. Representations of chordal bipartite graphs

Farber [5] has given an intersection graph characterization of strongly chordal graphs. Indoing so, he introduced the concept of compatible subtrees in a rooted weighted tree. Let T bea tree with a distinguished vertex, r , called the root of T . The tree T is weighted if each edgeof T is assigned a positive number, called the length of the edge. For u,v ∈ V (T ), the weighteddistance from u to v, denoted d∗

T (u, v), is the sum of the lengths of the edges on the uniquepath from u to v in T . Suppose that T1 and T2 are subtrees of T . We say that T1 succeeds T2 ifd∗T (r, u) > d∗

T (r, v) for all u ∈ V (T2)−V (T1) and v ∈ V (T1)∩V (T2). Equivalently, T1 succeedsT2 if, for all u,v ∈ V (T2) with d∗

T (r, u) � d∗T (r, v), v ∈ V (T1) implies u ∈ V (T1). The two trees

T1 and T2 are called compatible if either T1 succeeds T2 or T2 succeeds T1.Let G be a graph and v be a vertex of G. Then v is called a simple vertex of G if for any

u,u′ ∈ N [v], either N [u] ⊆ N [u′] or N [u′] ⊆ N [u]. When every induced subgraph of G has asimple vertex, G is called strongly chordal.

Since a simple vertex is simplicial, every strongly chordal graph is a chordal graph. It followsfrom the definition that every strongly chordal graph G admits a simple elimination ordering,that is, a vertex ordering v1, v2, . . . , vn of G such that for each i � 1, vi is a simple vertex of thesubgraph induced by {vi, vi+1, . . . , vn}. In fact, as shown in [5,6], strongly chordal graphs areprecisely the graphs which admit simple elimination orderings.

Theorem 3.1. [5] Let G be a graph with vertices v1, v2, . . . , vn. Then G is strongly chordal if andonly if there is a family of pairwise compatible subtrees Ti , i = 1,2, . . . , n, in a rooted weightedtree T , such that for 1 � i < j � n, vi and vj are adjacent if and only if V (Ti) ∩ V (Tj ) = ∅.

Let H be a bipartite graph with bipartition (X,Y ). A vertex v of H is simple if for any u,u′ ∈N(v) either N(u) ⊆ N(u′) or N(u′) ⊆ N(u). Suppose that L: v1, v2, . . . , vn is a vertex orderingof H . For each i � 1, we use Hi to denote the subgraph of H induced by {vi, vi+1, . . . , vn}.We say that L is a simple elimination ordering of H if vi is a simple vertex in Hi for eachi = 1,2, . . . , n.

Theorem 3.2. [15] Let H be a bipartite graph with bipartition (X,Y ). Then H is chordal bipar-tite if and only if it has a simple elimination ordering.

Furthermore, suppose that H is chordal bipartite. Then there is a simple elimination orderingy1, . . . , ym, x1, . . . , xn, where X = {x1, x2, . . . , xn} and Y = {y1, y2, . . . , ym}, such that if xi and


xk with i < k are both neighbours of some yj , then NH ′(xi) ⊆ NH ′(xk) where H ′ is the subgraphof H induced by {yj , . . . , ym, x1, . . . , xn}.

A hypercircle is a graph which consists of internally vertex disjoint paths between two distin-guished vertices. Let C∗ be a hypercircle with r , s being the two distinguished vertices. A clawin C∗ is a connected subgraph of C∗ containing exactly one of the two vertices r and s. Clearlyevery claw is a tree. Denote by T the set of all claws. Thus T can be partitioned into two sets Tr

and Ts , where Tr consists of all claws containing r and Ts consists of all claws containing s. Foreach claw T ∈ T , let T − denote C∗ − V (T ). Then T ∈ Tr if and only if T − ∈ Ts .

We again assume C∗ is weighted. Suppose that T1 and T2 are two claws in T . Then exactlyone of pairs {T1, T2}, {T1, T

−2 }, {T −

1 , T2}, {T −1 , T −

2 } is a subset of Tr and hence can be viewed asa pair of subtrees in the rooted weighted tree C∗ − s with root r . If the unique pair of subtrees ofC∗ − s is compatible, then we say that T1 and T2 are compatible with respect to r .

A graph G is a compatible claw intersection graph if there is a weighted hypercircle C∗with distinguished vertices r, s and a family of pairwise compatible claws Tv , v ∈ V (G), withrespect r , such that for any u,v ∈ V (G), uv ∈ E(G) if and only if V (Tu) ∩ V (Tv) = ∅.

Theorem 3.3. A bipartite graph is chordal bipartite if and only if the complement of H is acompatible claw intersection graph.

Proof. Let H be a bipartite graph with bipartition (X,Y ). Suppose that the complement of H

is a compatible claw intersection graph. Assume that H is connected. Otherwise we considerthe connected components of H separately and show that every connected component of H ischordal bipartite. Let C∗ be a weighted hypercircle with distinguished vertices r , s and let Tv ,v ∈ X∪Y , be a family of pairwise compatible claws with respect to r in C∗ such that two verticesu,v are adjacent in H if and only if V (Tu) ∩ V (Tv) = ∅. Suppose that there exist x′ ∈ X andy′ ∈ Y such that Tx′ and Ty′ both contain r . Let Ur ⊆ V (H) consists of all vertices whose clawscontain r and Us ⊆ V (H) consists of all vertices whose claws contain s. Since x′ ∈ Ur ∩ X andy′ ∈ Ur ∩ Y , we have (Ur ∩ X) ∪ (Us ∩ Y) = ∅ and (Ur ∩ Y) ∩ (Us ∩ X) = ∅. But H has noedges between any vertex of (Ur ∩ X) ∪ (Us ∩ Y) and any vertex of (Ur ∩ Y) ∩ (Us ∩ X). Thiscontradicts the assumption that H is connected. Hence if Tu and Tv both contain r then eitheru,v ∈ X or u,v ∈ Y ; similarly, if Tu and Tv both contain s then either u,v ∈ X or u,v ∈ Y . So bysymmetry we may assume that each Tx with x ∈ X contains s and each Ty with y ∈ Y contains r .To show that H is chordal bipartite it suffices to show that H contains a simple vertex.

Let f ∈ Y be such that Tf contains a vertex w whose distance from r is the largest amongall vertices in

⋃y∈Y V (Ty). That is, d∗

C∗−s(r,w) � d∗C∗−s(r, v) for all v ∈ ⋃

y∈Y V (Ty). Sup-pose that there are two vertices a, b ∈ NH (f ) such that NH (a) � NH (b) and NH (b) � NH (a).Let c ∈ NH (a) − NH (b) and d ∈ NH (b) − NH (a). Since b and c are not adjacent in H ,V (Tb) ∩ V (Tc) = ∅. Thus there exists a vertex u ∈ V (Tb) ∩ V (Tc). Similarly, there exists avertex v ∈ V (Ta) ∩ V (Td). Clearly, neither u nor v is in Tf , as V (Tf ) ∩ V (Ta) = ∅ andV (Tf ) ∩ V (Tb) = ∅. Since a and c are adjacent, V (Ta) ∩ V (Tc) = ∅, which implies that u ∈V (T −

a ). Similarly, v ∈ V (T −b ). It follows from the choice of f that d∗

C∗−s(r, u) � d∗C∗−s(r,w)

and d∗C∗−s(r, v) � d∗

C∗−s(r,w). Since w is not a vertex of Ta or of Tb , w is a vertex of T −a and

of T −b . Hence T −

a contains both vertices w and u, and T −b contains both w and v. By assumption

T −a and T −

b are compatible. So either T −a succeeds T −

b or T −b succeeds T −

a . But the definitionimplies that if T −

a succeeds T −b then v ∈ V (T −

a ), and if T −b succeeds T −

a then u ∈ V (T −b ). In

either case we obtain a contradiction. Therefore the vertex f is a simple vertex of H .


Conversely, suppose that H is chordal bipartite. If Y contains an isolated vertex, then wemove it from Y to X. So we may assume that every vertex of Y is adjacent to at least onevertex in X. By Theorem 3.2, there is a simple elimination ordering y1, . . . , ym, x1, . . . , xn

where X = {x1, x2, . . . , xn} and Y = {y1, y2, . . . , ym}. Furthermore, if xi, xk with i < k areboth neighbours of some yj , then NH ′(xi) ⊆ NH ′(xk) where H ′ is the subgraph of H in-duced by {yj , . . . , ym, x1, . . . , xn}. Let S be the tree with vertex set X ∪ Y ∪ {r} and edge set{rxi : 1 � i � n} ∪ {yjxi : 1 � j � m, i = min{k: xkyj ∈ E(H)}}. Let d∗

S(r, xi) = 1 for each i

with 1 � i � n, and d∗S(yj , xi) = m − j + 1 for each j with 1 � j � m. Then S can be viewed as

a rooted weighted tree with root r .We first construct a family of pairwise compatible subtrees Rv , v ∈ X ∪ Y , in S, each con-

taining the root r . For each j with 1 � j � m, let Ryjbe the unique path of S from r to yj .

For each i with 1 � i � n, let Rxibe the subtree of R induced by {r, x1, x2, . . . , xi} ∪ NH (xi).

It is easy to see from the definition that for any x ∈ X and y ∈ Y , xy ∈ E(H) if and only ifV (Rx) ⊇ V (Ry). For each j with 1 � j � m, since Ryj

is a path from r to yj , any subtreeof S containing r succeeds Ryj

. Thus we only need to show that Rxiand Rxk

are compatiblefor any 1 � i < k < n. If NH (xi) ∩ NH (xk) = ∅, then Rxk

contains every vertex of Rxiof dis-

tance at most one from r and contains no vertex of Rxiof distance greater than one. Hence

Rxksucceeds Rxi

and the two trees are compatible. So suppose that NH (xi) ∩ NH (xk) = ∅. Letyj be the common neighbour of xi and xk in H with the smallest subscript. By assumptionNH ′(xi) ⊆ NH ′(xk) where H ′ is the subgraph of H induced by {yj , . . . , ym, x1, . . . , xn}. Fromthe definition, d∗(r, yj ) = m − j + 2 < m − l + 2 = d∗(r, yl) for each l with l < j . Thus Rxk

contains all vertices of Rxiof distance � d∗(r, yj ) from r and contains no vertex of Rxi

of dis-tance > d∗(r, yj ) from r . Hence Rxk

succeeds Rxi. Therefore the subtrees Rv , v ∈ X ∪ Y , are

pairwise compatible.We next modify S to a new weighted tree which consists of internally vertex disjoint paths

from the vertex r . So suppose that the neighbours of xi in S are r, yj1 , yj2, . . . , yjq . We deletexi from R and then add q new vertices xi1, xi2, . . . , xiq , and new edges rxi1, rxi2, . . . , rxiq

and xi1yj1, xi2yj2, . . . , xiq yjq . For each k = 1,2, . . . , q , assign length one to rxik and lengthm − jk + 1 to xikyjk

. Note that each y ∈ Y has the same distance from r in R as it has in S.Denote by R the resulting weighted graph. It is easy to see that R consists of paths from r , whichis again treated as the root of R. We now accordingly modify each Rv , v ∈ X ∪ Y , in S, so that itbecomes a subtree Sv of R. Each Sv must contain r . If Rv contains some xi in S, then Sv containsall the replaced vertices of xi in R and if Rv contains any yj in S, then Sv must contain yj in R.Let Sv contain no other vertices. Thus we obtained a family of subtrees Sv , v ∈ X ∪ Y , in R,each containing r . It is easy to see that the subtrees Sv in R still maintain the property of beingpairwise compatible and the property that for any x ∈ Y , y ∈ Y , x and y are adjacent if and onlyif V (Sx) ⊇ V (Sy).

Finally, we show how to construct a weighted hypercircle C∗ and a family of compatible clawsin C∗, whose intersection graph is the complements of H . Let C∗ be obtained from R by addinga new vertex s and letting s be adjacent to each leaf of R. All edges incident with s are of lengthone in C∗. Clearly, C∗ is a weighted hypercircle with r , s being the two distinguished vertices.All subtrees Sv , v ∈ X ∪ Y , can now be treated as claws in C∗. We define a family of claws Tv ,v ∈ X ∪Y , in C∗ as follows: For each x ∈ X, let Tx = S−

x , and for each y ∈ Y , let Ty = Sy . Theseclaws are pairwise compatible with respect to r as verified above. For any x, x′ ∈ X, V (Tx) ∩V (Tx′) = ∅, as both Tx and Tx′ contain s. Similarly, for any y, y′ ∈ Y , V (Ty) ∩ V (Ty′) = ∅,as both Ty and Ty′ contain r . For any x ∈ X and y ∈ Y , x and y are adjacent if and only if


V (Sx) ⊇ V (Sy), i.e., V (Tx) ∩ V (Ty) = ∅. Therefore the complement of H is a compatible clawintersection graph. �

We remark that Theorem 3.3 remains valid if each hypercircle is restricted to satisfy the prop-erty that all paths between the two distinguished vertices have the same total (weighted) length.

A bipartite graph H with bipartition (X,Y ) is called a compatible subtree containment bi-graph if there is a family of pairwise compatible subtrees Sv , v ∈ X ∪ Y , in a rooted weightedtrees, such that for any x ∈ X and y ∈ Y , x and y are adjacent if and only if V (Sx) ⊇ V (Sy).It follows from Theorem 3.3 that every chordal bipartite graph is a compatible subtree contain-ment bigraph. In fact, the converse is also true; the existence of a simple vertex can be shown byextracting the first part (i.e., the second paragraph) of the proof of Theorem 3.3. Hence chordalbipartite graphs are precisely the compatible subtree containment bigraphs.

Let H be a bipartite graph with bipartition (X,Y ). Following [1], we use splitX(H) to denotethe graph obtained from H by completing X to a clique. Suppose that L: v1, v2, . . . , vn is asimple elimination ordering of H , in which all vertices of X appear after each vertex of Y . ThenL is a simple elimination ordering of splitX(H) and hence splitX(H) is strongly chordal.

A bipartite graph H with bipartition (X,Y ) is called a compatible subtree intersection bigraphif there exists a family of pairwise compatible subtrees Tv, v ∈ V (H), in a rooted weighted treesuch that for any x ∈ X and y ∈ Y , x and y are adjacent if and only if V (Ty) ∩ V (Tx) = ∅.

Theorem 3.4. A bipartite graph is chordal bipartite if and only if it is a compatible subtreeintersection bigraph.

Proof. Let H be a bipartite graph with bipartition (X,Y ). Suppose that H is chordal bipartite.Then by Theorem 3.2 H admits a simple elimination ordering v1, v2, . . . , vn, in which all verticesof X appear after each vertex of Y . Hence splitX(H) is strongly chordal. By Theorem 3.1 thereis a family of pairwise compatible subtrees Ti , i = 1,2, . . . , n, in a rooted weighted tree T suchthat, for 1 � i < j � n, vi and vj are adjacent in splitX(H) if and only if V (Ti) ∩ V (Tj ) = ∅.Since the only edges of splitX(H) not in H are between vertices of X. So for any vi ∈ X andvj ∈ Y , vi and vj are adjacent if and only if V (Ti) ∩ V (Tj ) = ∅.

Conversely, suppose that there is a family of pairwise compatible subtrees Ti , i = 1,2, . . . , n,in a rooted weighted tree T such that vi ∈ X and vj ∈ Y are adjacent if and only if V (Ti) ∩V (Tj ) = ∅. Let G be obtained from H by adding all missing edges vivj such that V (Ti) ∩V (Tj ) = ∅. Then by Theorem 3.1 G is strongly chordal and hence admits a simple eliminationordering. Since each added edge in G is either between some two vertices of X or between sometwo vertices of Y , the same vertex ordering of G is a simple elimination ordering of H . Thereforeby Theorem 3.2 H is chordal bipartite. �

Combining Theorems 3.3 and 3.4, and the comment following the proof of Theorem 3.3, wehave:

Corollary 3.5. The following statements are equivalent for a bipartite graph H :

(i) H is chordal bipartite;(ii) H is a compatible subtree intersection bigraph;

(iii) H is a compatible subtree containment bigraph;(iv) the complement of H is a compatible claw intersection graph.


As a final remark, a bipartite graph H is of Ferrers dimension at most k if and only if thecomplement of H is the intersection graph of a family of (not necessarily compatible) claws ina hypercircle, which consists of k internally vertex disjoint paths between two distinguished ver-tices. This can be shown by combining Theorem 2.2 and a proof similar to that of Theorem 2.3.When a hypercircle consists of exactly two internally vertex disjoint paths between two distin-guished vertices (i.e., a circle with two distinguished vertices), any two claws in the hypercircleare compatible. So the complements of bipartite graphs of Ferrers dimension at most two areprecisely the intersection graphs of claws in circles (see Theorem 2.3).

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Representation characterizations of chordal bipartite graphs