INVERTING SIGNED GRAPHS* 1. Introduction. In

SIAM J. ALG. DISC METH. Vol. 5, No. 2, June 1984

@ 1984 Society for Industrial and Applied Mathematics

007

INVERTING SIGNED GRAPHS* HARVEY J. GREENBERG,? J. RICHARD LUNDGRENS

AND

JOHN S. MAYBEEB

Abstract. This paper addresses the question of determining the class of rectangular matrices having a given signed graph as a signed row or column graph. We also determine equivalent conditions on a given pair of signed graphs in order for them to be the signed row and column graphs of some rectangular matrix. In connection with these signed graph inversion problems we discuss the concept of minimality and illustrate how to invert a pair of signed graphs.

1. Introduction. In this paper we continue the systematic investigation of the structural relationships between rectangular matrices and graphs, digraphs and signed graphs started in [4], [5], and [6]. To study these relationships we make use of the following graphs. Given an m X n matrix A, we define two sets of points R ={r,, . . . , r,,,) and C = {c,, . . , c,) to represent the rows and columns of A, respectively. The three basic graphs are: Fundamental bigraph. BG is a bipartite graph (bigraph) on R and C. The lines correspond to the nonzeros of A; i.e., [r,, c,] is a line in BG if and only if a,, # 0. Row graph. RG has point set R. The line [r,, rk] belcngs to RG if there exists c, E C such that [r,, cj] and [rk,c,] are lines of BG. Column graph. CG has point set C. The line [c,, ck] belongs to CG if there exists r, E R such that [c,, r,] and [ck,r,] are lines of BG. This leads naturally to the questions of determining the class of rectangular matrices having a given graph as a row or column graph and determining equivalent conditions on a given pair of graphs in order for them to be the row and column graphs of some rectangular matrix (see [7]). These graph inversion techniques are useful in characterizing the two-step graphs studied by Exoo and Harary [2] (see [S]) and in characterizing the competition graphs studied by Roberts [16], [17] (see [14]). In this paper we turn our attention to these same problems for signed graphs. First, we consider how the sign information in the matrix can be incorporated into the three graphs. It is clear that the sign information in the real matrix A can be immediately incorporated into the bigraph BG. In fact, we label the line [r,, c,] positive if a,, > 0 and negative if a, < 0. The resulting signed graph will be denoted BG+. The signed structure of A, i.e., the locations of the positive and negative entries of A, is immediately discernible from the signed bigraph BG'. Thus, given a signed bigraph G', we can construct a unique matrix A with entries +1, -1 or 0 such that BG'(A) = G+. Now it is not always possible to form signed row or column graphs. To form RG+(A),it is necessary that the scalar product of any two rows be positive, negative or zero independently of the magnitudes of the elements; i.e., all terms in the scalar product are weakly of the same sign. We can then form RG+ where the line [r,, r,] is positive if the corresponding row vectors have a positive scalar product, and negative if the scalar product is negative. CG+ is defined in a similar way. In [6] it was shown that RG+ can be formed if and only if CG+ can be formed, and if so, we say that A is signed. Applications of these signed graphs and the importance of when they can be formed are discussed in Greenberg [3], Greenberg, Lundgren and Maybee [6],

.

*Received by the editors June 15, 1982, and in revised form May 3, 1983. t Energy Information Administration, Washington, DC 20461. $University of Colorado at Denver, Denver, Colorado 80202. 8 University of Colorado, Boulder, Colorado 80309.

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Kydes and Provan [13] and Provan [15]. These include identifying and characterizing important components of energy economic models such as physical flows matrices and transportation matrices, and analyzing correlation and determinacy in linear systems related to networks. In [9] we show how the signed graph inversion method developed in this paper can be used to teach a computer to build models using partial information in the form of economic correlation. In § 2 we find the class of matrices A satisfying R G + ( A ) = G + (and CG+(A)= G+) for a given signed graph G+. We use the methods developed in [7] as well as a theorem of Harary and Kabell [ I 11 on marked graphs. We also discuss the notion of minimality. In § 3 we find necessary and sufficient conditions for a pair of signed graphs to be invertible and illustrate how to invert a pair of signed graphs.

2. One-graph inversion. In [7] we characterized the family of regular Boolean matrices A whose row (column) graph equals a specified graph G. ( A Boolean matrix is regular if each row and column has a nonzero entry.) Here we investigate the same problem for signed graphs and regular signed Boolean matrices (entries are *1 or 0, and each row and column has a nonzero entry). As in [7], observe that if RG+(A) = G+, then c G + ( A ~ )= G+. Consequently, we shall consider only matrices A such that RG+(A) = G+. Before considering the signed case, we review the situation for graphs. A k-clique, k 2 1, of a graph is a complete subgraph on k points. Given a graph G, a finite set S of cliques of G will be called a clique cover if every point and line of G belongs to at least one clique in S. We will use the notation (X) to denote the subgraph of G generated by the set of points X. The following result is [7, Thm. 11. THEOREM2.1. Given the graph G = ( V, E) with p = I VI, the regular Boolean matrix A has the property that R G ( A ) = G if and only if A hasp rows and the columns of A correspond to a clique cover of G. To illustrate the difference between the two problems, we consider the following example. Let G and G + be as illustrated in Fig. 2.1, where we have followed the convention of using dashed lines to represent negative lines, as introduced in [12].

Then S = {(1,2,4), (2,3,4)) is a clique cover for G. So by Theorem 2.1,

satisfies R G ( A ) = G. However, even though S is also a clique cover for G+, there is

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no way to sign A so that RG'(A) = G+. Now, if we let T = {(1,2,4)(2,3)(3,4)}, then

also satisfies RG(A) = G. Here again, T is a clique cover of G C also, but if we sign A to get

then RGC(A,)= G+. The difference in the two clique covers is that each clique in T is balanced. A signed graph G + is balanced if and only if the points of G+ can be partitioned into disjoint subsets S1 and S2 (one of which may be empty) such that every line joining two points in the same set is positive a11d every line joining two points in different sets is negative. Given a signed graph GC,a finite set S+ of signed cliques of G + will be called a balanced clique cover of G + if every point and line of G+ belongs to at least one clique in SC,and every clique in SCis balanced. The next two lemmas show that every signed graph G + has a balanced clique cover which can be used to construct a matrix A satisfying RGC(A)= G+. LEMMA2.2. Every signed graph G+ has a balanced clique cover. Proof Let SCbe the set of all lines in G+ together with the isolated points. Clearly SC is a balanced clique cover of G+. 0 LEMMA2.3. Let G C= ( V, E ) be a signed graph with p = I VI, n = I E ~ , and po equal to the number of isolated points in G+. Then there exists a p X ( n + p o ) regular signed Boolean matrix A such that RG'(A) = GC. Proof Label the points of G+ 1 , 2 , ,p and the lines 1 , 2 , . . . , n. We then construct A as follows. For each line of G Cthere is a corresponding column of A with 1's in rows i and j if the line [ i , j ] is positive, and 1 in row i and -1 in row j if the line is negative and i < j. For each isolated point k of G+ there is a corresponding column of A with a 1 in row k. Clearly A satisfies the conditions of the lemma. 0 The columns of matrix A constructed in Lemma 2.3 corresponded to the cliques in the balanced clique cover consisting of all the lines and isolated points. However, for the graph G' in Fig. 2.1, we found a matrix A satisfying RG+(A) = G+ where the cliques determined by the columns of A were not all lines or points. To construct a matrix A corresponding to an arbitrary balanced clique cover of G + ,we need a method for determining the signs in each column of A. For this we use the notion of marked graphs investigated by Bieneke and Harary [I], Harary [lo], and Harary and Kabell [ I l l-. In a marked graph, the points are designated positive or negative. Let M be a marked graph with underlying graph G = G(M) having the same points and lines as M, but without any signs on its lines or points. The signed graph of the marked graph M, written S(M), is obtained from G by affixing to each line the product of the signs of its two points. The following result is [ l o , Thm 3.81 (also see Harary and Kabell [ I 11).

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THEOREM2.4. To each marked graph M there corresponds a unique balanced signed graph B'= S ( M ) . To each balanced signed graph B+ there correspond two marked graphs M and M ' , which are sign-reversals, such that S ( M )= S ( M 1 )= B'. Given a balanced signed graph B', we construct a marked graph M satisfying S ( M )= B' as follows. Select an arbitrary point and mark it positive (or negative). Select a point adjacent to this point and label it with the product of the sign of the marked point and the sign of the line joining the two points. Continue in this way until the graph is marked. We can use this procedure to see how the matrix A, was constructed from the graph G+ of Fig. 2.1. Since T is a balanced clique cover of G', we can mark each of the cliques in T independently as follows:

i 5 ,

(Z,3),

(54).

A, is then formed using the signs for each clique. This procedure can be used for any balanced clique cover to get the following theorem. THEOREM2.5. Given the graph G f = ( V ,E ) with p = I VI, the regular signed Boolean matrix A has the property that R G ' ( A ) = G+ if and only if A hasp rows and the columns of A correspond to a balanced clique cover of G+. Proof. Suppose R G + ( A )= G+. Then by Theorem 2.1, the columns of A correspond to a clique cover of G + ,and by [6, Lemma 31, each of these cliques is balanced. For the converse, let A be a regular signed Boolean matrix with p rows whose columns correspond to a balanced clique cover of G'. Observe that we can form R G + ( A ) ,since if r, and r, have nonzeros in columns k and q, then [r,, r,] is in cliques Ck and C,, and so r, and r, are marked with either the same signs or opposite signs in both Ck and C,. We must show that R G + ( A )= G+. Now [r,, r,] is negative in G' if and only if [r,,r,] belongs to a balanced clique, and r, and r, are marked with opposite signs if and only if a column of A contains nonzero entries of opposite signs in rows r, and r, if and only if [r,, r,] is negative in R G + ( A ) .Similarly, [r,,r,] is positive in G' if and only if [r,,r,] is positive in R G + ( A ) . Hence, R G + ( A )= G + , and the proof is complete. 0 Given a balanced clique cover S+ of the signed graph G+, we can form a signed clique cover graph, Q ( S + ) as , follows. Let S+ = {C,, . . , C,,) and A be the corresponding matrix. Then Q ( S + )is a signed graph on the points 1, 2 , . . . , n, and the line [i,j] is positive in Q ( S + )if and only if C, and C, contain at least one point marked with the same sign, and negative if and only if C, and C,contain at least one point marked with opposite signs. Observe that if Ci and C j contain more than one common point, they are either all marked with the same sign or all marked with opposite signs, since the determination of the signs in Q ( S + )corresponds to the determination of the signs in C G + ( A ) ,which can be formed since A is signed. In fact, this construction shows that C G + ( A )= Q + ( S ) .Hence, we can reformulate the above theorem as follows. THEOREM2.6. Given the graph G+= ( V, E) with p = I VI, the regular Boolean matrix A has the property that R G + ( A )= G if and only if A hasp rows and there exists a balanced clique cover S+ of G+ such that C G + ( A )is isomorphic to Q(S+). The above relationship between signed graphs and rectangular matrices leads to the following graph theoretic result. THEOREM2.7. Let G + be a signed graph and S+ a balanced clique cover of G+. Then Q ( S + ) is balanced if and only if G + is balanced. Proof. Let A be the matrix constructed as described in the comments following Theorem 2.4. Then C G + ( A )is balanced if and only if R G + ( A ) is balanced by [6, Thm. 31. Hence, the result follows since Q ( S + )= C G + ( A )and R G ' ( A ) = G+. 0

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Now let R (G+)= {A :A is a regular signed Boolean matrix and RG+(A) = G+). As in 171, we can consider the notion of minimality for the set R(G+). That is, given a matrix A E R ( G + ) , it is frequently important to find a matrix A ' E R ( G + ) that has fewer nonzeros than A or fewer columns than A. Basically, the same results hold as in 171, so we will not develop the theory of minimality here. However, there are two significant differences that we will describe. First, for a graph G, the minimum number of columns for a matrix A in R ( G ) was determined by k ( G ) , the clique cover number of G. However, as is illustrated by the graphs in Fig. 2.1, a clique cover of G + with the smallest number of cliques may not be a balanced clique cover. To find a matrix A in R ( G + ) with the minimum number of columns, we must find a balanced clique cover with the smallest number of cliques. This number is denoted by k+(G+).Clearly, k(G+) Z k+(G+). The other difference is the number of matrices A in R ( G + ) corresponding to a particular balanced clique cover S+ = {C,, . . , C,,) Since to each Ci there correspond two marked graphs M, and MI, the above construction leads to 2" p X n matrices A with RG+(A) = G + and CG+(A)= Q(S+). This situation is illustrated in Fig. 2.3 for the graph G + and balanced clique cover S+ in Fig. 2.2. Observe that since CG+(A)= CG+(-A), there are at most 2"-' different graphs Q(S+) corresponding to the different ways of marking the cliques. Since RG+(A) is balanced, then CG+(A) must be balanced. Since CG(A) is a 3-clique, there are only two ways of signing a 3-clique so that it is balanced. Hence, in this case, there are only two nonisomorphic signed clique cover graphs but four ways of signing the particular graph so that it is balanced.

Remark. It appears that what happens in the above example also happens in general. That is, let G + be balanced and S+ be a balanced clique cover of G+. Then all possible ways of signing Q(S+) so that it is balanced can be realized by changing the marked graphs for the various cliques in S+.

3. Two-graph inversion. In this section we consider the problem of inverting a pair of signed graphs. That is, given signed graphs G: and G l , when can a regular signed Boolean matrix A be constructed having the property that RG+(A) = G: and CG+(A)= G:? If such a matrix A exists, as in [7], we say that G: and G: are invertible. Observe that for a regular matrix A, the signed graphs R G + ( A )and CG+(A) have the same number of components by [5, Cor. 2.31. It follows that we cannot always

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invert the pair G:, G:. However, we can provide equivalent conditions for existence analogous to those given for a pair of graphs in [7, Thm. 21. THEOREM 3.1. Given two signed graphs G: and G:, the following are equivalent: (i) G: and G: are invertible; (ii) G: is isomorphic to a signed clique cover graph of G:; (iii) G: is isomorphic to a signed clique cover graph of G:. The proof is essentially the same as the proof of [7, Thm. 21, except that Theorem 2.6 is used instead of the one-graph inversion theorem of [7]. Now we will illustrate how Theorem 3.1 can be used to invert a pair of signed graphs. Consider the pair of graphs in Fig. 3.1. If we choose the balanced clique cover + + -

s = {(I, 4,5), (i,3,a), (3,3,a)}

and mark each clique in S as shown,,then G: is isomorphic to Q(S+) as illustrated in Fig. 2.3. Using the signs for each clique, we can then construct

Clearly, RG+(A)= G: and CG+(A)= G:. Observe that not only did we have to find a balanced clique cover S, but that we also needed an appropriate marking of the cliques in order to get Q(S+)= G:. We close this section by observing that the concept of minimality can be developed in the same way as in [7].

REFERENCES [I] L. W. BEINEKEAND F. HARARY, Consistency in marked digraphs, J . Math. Psych., 18 (1978), pp. 260-269. [2] G. E x 0 0 AND F. H A R A R Y ,Step graphs, J. Combin., Inform. System Sci., (1984), to appear.

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[3] H. J. GREENBERG, Measuring complementarityandqualitative determinacy in matnctal forms, proceedings of the Symposium on Computer Assisted Analysis and Model Simplification, H. J. Greenberg and J. S. Maybee, eds, Academic Press, New York, 1981. [4] H. J. GREENBERG,J. R. LUNDGRENAND J. S. MAYBEE,Graph theoretic foundations of computerassisted analysis, in Proceedings of the Symposium on Computer Assisted Analysis and Model Simplification, H. J. Greenberg and J. S. Maybee, eds., Academic Press, New York, 1981. [51 , Green theoretic methods for the qualitative analysis of rectangular matrices, this Journal, 2 (1981), pp. 227-239. [61 , Rectangular matrices and signed graphs, this Journal, 4 (1983), pp. 50-61. [71 , Inverting graphs of rectangular matrices Discrete Applied Math., (1984), to appear. [a] , The inversion of 2-step graphs. . J. Combin., Inform. System Sci., (1984), to appear. P I -, Signed graphs of netforms, (1984), to appear. [lo] F. HARARY,Structural models and graph theory, in Proceedings of the Symposium on Computer Assisted Analysis and Model Simplification, H. J. Greenberg and J. S. Maybee, eds., Academic Press, New York, 1981. [ l l ] F. HARARYA N D J. A. KABELL,A n eficientalgorithm to detect balance in signedgraphs, Mathematical Social Sciences, 1 (1980). [12] F. HARARY,R. Z. NORMANAND D. CARTWRIGHT,Structural Models: A n Introduction to the Theory of Directed Graphs, John Wiley, New York, 1965. [13] A. KYDES A N D J. S. PROVAN,Correlation and determinacy in network models, BNL Report 51243, Brookhaven National Laboratory, Upton, New York, 1980. [14] J. R. LUNDGRENAND J. S. MAYBEE,A characterization of graphs of competition number m, Discrete Applied Math., 6 (1983), pp. 319-322. [IS] J. S. PROVAN,Determinacy in linear systems and networks, this Journal, 4 (1983), 262-278. [16] F. S. ROBERTS, Food webs, competition graphs, and the boxicity of ecological phase space, Theory and Applications of Graphs-in America's Bicentennial Year, Y. Alavi and D. Lick, ed., SpringerVerlag, New York, 1978. ~171 , Graph Theory and Its Applications to Problems of Society, CBMS Regional Conference Series in Applied Mathematics 29, Society for Industrial and Applied Mathematics, Philadelphia, 1978.

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