ALGORITHMIC ASPECTS OF BIPARTITE GRAPHS

We generalize previous work done by Donald J. Rose and Robert E. Tarjan [2], who developed efficient algorithms for use on directed graphs. This paper considers an edge elimination process on bipartite graphs, presenting several theorems which lead to an algorithm for computing the minimal fill-in of a given ordered graph.


INTRODUCTION
Gaussian elimination is commonly used in matrix theory.If the matrix is sparse, that is, contains a large number of zero entries, it is advantageous to avoid unnecessary operations on those zeros in order to save memory and run time.When we perforn Gaussian elinination, some of the zeros in the matrix may be replaced by nonzeros.These new nonzeros are called "fill-in".A perfect ordering is a sequence of Gaussian pivots which produces no fill-in.
Much in known about the case in which the coefficient matr;x is symmetric (see Golumbic [1]).It is also known how to determine fill-ins for nonsymmetric matrices when the pivots are chosen on the main diagonal.Algorithms have been developed for computing the fill-in for any ordering of pivots, for generating a perfect ordering if such an ordering exists, and for reducing a fill-in to a minimal one (Rose and Tarjan [2]).However, it may be the case that choosing pivots which are not on the main diagonal produces a smaller fill-in than does choosing pivots which are on the main diagonal.For example, choosing any pivot on the nain diagonal in the following matrix produces a fill-in of at least one entry.(The entries denoted by an X represent non-zeros in the matrix.)X X X 0 0 X X O/ 0 0 X X But choosing the entry in row two, column one produces no fill-in, and, in fact, there is a perfect elimination ordering for this matrix if pivots off the main diagonal are chosen.
Therefore, this paper generalizes the results of Rose and Tarjan [2] in order to develop an algorithm which takes all nonzero entries of a given matrix into account, as opposed to only the entries on the main diagonal.We draw heavily from several chapters of a book by Golumbic [1], and some of the proofs of Rose and Tarjan [2] can be readily adapted for our purposes.

DEFINITIONS AND NOTATION
We generally follow the same notation as Golumbic [1] and Rose et al. [3].
An undirected graph is a pair G (V,E), where V is a finite set of n IV[ elements called vertices and E C_ {(v,w)lv,w E V,v w} is a set of e [E unordered vertex pairs called edges.An undirected graph G (V,E) is bipartite if its vertices can be partitioned into two disjoint sets V X + Y, i.e. every edge has one endpoifft in X and the other in Y.We use the notation H (X,Y,E) to denote bipartite graphs.A bipartite graph H (X,Y,E) is complete if for every x E X and y E Y we have (x,y) E.
{w Vl(v,w  E} is the set of vertices adjacent to v.For a bipartite graph I-I and (x,y) e E, Adj(x,y) {{(x,k)lk Y} U {(J,y)lJ e X}) is the set of edges adjacent to (x,y).
A graph called chordal if every cycle of length strictly greater than 3 possesses a chord, i.e. an edge joining two nonconsecutive vertices of the cycle.Similarly, a bipartite graph is said to be btpartite chordal if every cycle of length strictly greater than four contains a chord.Given a subset A C_ V of the vertices, define the subgraph induced by A to be GA (A,EA)   where Ea {(x,y)lx A and y E A}.A subset A C_ V is called a clique if it induces a complete subgraph.We call a vertex x szmplicialif Adj(x) induces a complete subgraph.In a bipartite graph, a subset A C X t2 Y is called a biclique if it induces a complete bipartite subgraph.An edge e (x,y) of a bipartite graph H is called bisimplicial if Adj(x)+ Adj(y) induces a complete bipartite subgraph of H.
Denote by S, the set ofendpoints of the edges e,...,e, and let So q).We say that is a perfect edge elimination scheme for H if each edge e, is bisimplicial in the remaining induced subgraph Hv_s,_, and Hv-s.has no edge.Thus we regard the elimination of an edge as the removal of all edges adjacent to e.A graph with a perfect edge elimination ordering is called a perfect edge elimination graph.Since this paper deals primarily with edge elimination orderings, we will refer to edge elimination orderings as simply "elimination orderings" or "elimination schemes".
Finally, for a vertex v, the deficiency D(v) is the set of pairs defined by D(v) {(x, Adj(x), v Adj(y),x _ Adj(y),x y}.So an alternate definition for a vertex v to be simplicial is if D(v)= For an edge (x,y), the deficiency D(x,y) is the set of pairs defined by D(x,y) {(a,b)[ a,b Adj(x) + Adj(y), (a, b) E}.An alternate definition for an edge (x,y) in the bipartite graph H to be bisimplicial is if D(x,y) Given an arbitrary n n matrix M (mi), the bipartite graph H (X,Y,E), with X {x, ,x,,} and Y {y,...,y,}, and the property that m,i 0 whenever (x,,y,) E is called a bipartite graph of M. When reducing M by Gaussian elimination, the edges of D(xi, Y1 correspond exactly to the fill-in entries created when element mii is used as a pivot in the matrix M. Pivoting on m, is equivalent to making Adj(x) + Adj(yi) into a complete subgraph by adding any missing edges and deleting (xi,y:).Therefore, a graph which has a perfect scheme is equivalent to a matrix which can be reduced to the identity without fill-in.
Let M be a matrix with biapartite garph H and an ordering of H. Define F(H,) to be the set of all fill-ins generated by reducing M by Gaussian elimination when the entries corresponding to the ordering are succesively choosen to act as pivots.Then is a minimal elimination ordering of H if no other ordering satisfies F(I-I) C F(H).The ordering is a minimum elimination ordering of H if no other ordering $ satisfies IF(H)I < IF(H,)I.This paper deals with minimal orderings only.It is known that the computation of the minimum fill-in of a graph in NP-complete, Yannakakis [4].
3. CHARACTERISTICS OF BIPARTITE GRAPHS In the case of symmetric matrices, Gaussian elimination is modelled by vertex elimination in- stead of edge elimination.It is known that a symmetric matrix M has a perfect vertex elimination scheme if and only if its associated graph is chordal (Golumbic, [1]).But there is no character- ization of graphs with edge elimination schemes in terms of some forbiddden subgraphs.While it is true that every chordal bipartite graph is a perfect elimination bipartite graph (Oolumbic, [1]), the converse of this statement is false.That is, it may be the case that a perfect elimination bipartite graph is not chordal bipartite.In this paper, there will be no restriction that the graph must be bipartite chordal.
The bisimpliciality of an edge is a hereditary property.That is, if an edge is bisimplicial, it is also bisimplicialin an induced subgraph.In order to construct perfect schemes, it is possible to choose an arbitrary bisimplicial edge, remove it, and then repeat the process in the induced subgraph.A theorem from Golumbic [1] formalizes this observation.THEOREM 1. Ire (x,y) is a bisimplicial edge of a perfect elimination bipartite graph H (X,Y,E), then Hx_{}+y_{} is also a perfect elimination bipartite graph.
In other words, if H is a perfect elimination bipartite graph and (x,y) is any edge with D(x,y) 0, then there is a perfect elimination ordering with (1)= (x,y).
The following theorems, then, are generalizations from Rose and Tarjan [2].Theorem 2 is mentioned as Ex.4,p.285 in Golumbic [1] but for the sake of completeness it is proven here.
Theorem 2 essentially says that we can add the deficiency of an edge to a perfect elimination graph, and new graph will have the same perfect elimination scheme.THEOREM 2. Let H (X, Y, E) be a bipartite graph with perfect edge elimination scheme [(x,,y),(z2,y2),...,(z,,y,,)].Let e E E and let H'= (X,Y,E UD(e)).Then is a perfect edge elimination scheme for H'.
PtOOF.We must show that for any (x,,yk), (xk,y,,) e E t.JD(e) with m,p > k, (x,,y,) E t.J D(e).There are three cases to consider.In the following illustrations, the dotted lines indicate an edge in D(e), while solid lines indicate edges in E U D(e).
If > k, then (x, y,) E E, since is a perfect ordering for H.But then we have (x,, y,,,) E, and (z,,yq) E, which implies (x,,y,,) E U D(e). ]f < k then (x,y) EUI:}(e) which implies (x,y,) E. But in that case we return to Case 1, and (zp, y,) E E U D(e).
PROOF.By Theorem 2, the graph H' (X,Y,E U D(x,y)) has a perfect edge elimination scheme, and (x,y) is bisimplicial in H'.Then, Theorem implies that H(x,y) is also a perfect edge elimination graph.
The next theorem provides a characterisation of bipartite graphs with nonminimal fill-in, a theorem which is utilized extensively in the algorithm at the end of this paper.
THEOREM 3. Let H (X,Y,E) be a bipartite graph and let F be a fill-in for H.If F is not minimal, then there exists an edge e (x,y) such that D(x,y) C F and D'(x,y) {, where D'(x,y) is the deficiency of e in H' (X, Y,E U F). PROOF.Let S {(x,y) EE E O F D'(x,y) }.For any edge (x,y), D(x,y) C_ D'(x,y), so D(x,y) \ F C_ D'(x,y).Hence if (x,y) is in S, then D(x,y) \ F C_ }, or D(x,y) C_ F. Since H' is a perfect elimination graph, S }.Let (w,z) S. If D(w,z) C F, then the theorem holds with (x,y)= (w,z).
Suppose, on the other hand, that D(w,z) F. Let F0 be a fill-in for H such that F0 C F, and let be a perfect elimination ordering for H0 (X,Y,E U F0). Then D((1)) C F0 C F. By Theorem 2, is a perfect ordering for H' (X,Y,E U D(w,z)).Thus D'((1)) }.Hence the theorem holds with (x,y)= (1).
The next theorem states that given a bipartite graph with a perfect scheme and a supergraph with the same property, we can remove an edge from the supergraph such that the resulting graph is also a perfect elimination graph.THEOREM 4. Let H (X,Y,E) be a perfect elimination bipartite graph.If H' (X,Y, EUJ) and J }, and Ial >_ 2, then there exists an fE J such that H" (X,Y,EUJ\{f}) is also a perfect elimination graph.PROOF.Proof is by induction on n IEI.If n _< 1, the result is obvious.Suppose the result is true for all n _< no and let n no + 1.By Theorem 3, there exists an edge e (x,y) such that D(x,y) C J and D'(x,y) }.We have two cases.
Case 1: There is an edge el J which has x or y as an endpoint.By Theorem 2, there exists a perfect elimination ordering for H' with (1) (x,y), but this ordering is also perfect for H'= (X,Y,E \ Case 2: Case does not apply.Then by Theorem 3, since F is not minimal, there exists an edge e (x,y)such that D'(x,y) q)and D(x,y) C J. Then H(=,,) (X \ (x},Y \ {y},Ex\{=}+y\{y I.J D(x,y)) and Hi=.y (X \ {z},Y \ {y},Ex\{=}+y\{y (J J) are perfect elimination graphs.

MINIMAL FILL PROCEDURE
The previous theorems allow us to construct an efficient algorithm for determining the mini- mal fill-in of a given bipartite graph.The algorithm is most easily stated as a recursive procedure.
By Theorem 3, if the fill-in for a graph is not minimal, we can find an edge (x,y) whose defi- ciency in H is not aI1 of F and whose deficiency in H' is empty.If some edge e in F is adjacent to (x,y), we can delete e from F and repeat the procedure.Otherwise, we form the graph HI (X \ {z},Y \ {y},Ex_{=}+y_{} (3D(x,y)), which has a fill-in of F-D(x,y).The proce- dure is applied recursively so that F D(x,y) is reduced to a minimal fill-in FI in HI.We then form the graph H2 (X,Y,E t.J FI), which has fill-in D(x,y).Again, we apply the procedure recursively to reduce D(x,y) to a minimal fill-in F2 in H. Then F0 F t5 F is a minimal fill-in for H.The following algorithm summarizes this procedure.
Procedure MINFILL (X,Y,E,F,F0) BEGIN declare F,Fa set variables local to procedure MINFILL IF H (X,Y,E) has no edge (x,y) with D'(x,y) q) and D(xy) C F THEN F0 := F. ELSE BEGIN Let (x,y) be an edge with S'(x,y) q) and D(x,y) C F. IF F contains some edge e incident to (x,y) THEN (Call 1) MINFILL (X,Y,E,F-{e},F0) ELSE BEGIN (Call 2) MINFILL (X \ {z},Y \ {y},Ex\{.}+r\{,}U D(x,y),F D(x,y),F) (Call 3) MINFILL (X,Y,E Fx,D(x,y),F) Fo := F U F END END END THEOREM 5.If F is a fill-in for a graph H (X,Y,E), then the set Fo computed by the execution of MF(X,Y,E,F,F0) is a minimal fill-in contained in F.
PROOF.The proof largely follows that of Rose and Tarjan [2].We proceed by induction on the number of edges in F. If F }, the theorem is obviously true.Suppose the theorem is true IFI < k =d Now, if H (X,Y,E) has no edge (x,y) such that D'(x,y) ) and D(x,y) C F, then If F contains an edge e incident to (x, y), then F {e} is a fill-in by the proof of Theorem 4. Then by the induction hypothesis, the set F0 C_ F e} computed by the recursive call (Call 1) is a minimal fill-in.

F0F
is minimal by Theorem 3.So suppose (x,y) is an edge satisfying D'(x,y) { and D(x,y) C F.