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A Remark on Perfect Gaussian Elimination of Symmetric Matrices
Europ. J. Combinatorics (1988) 9, 547-549
A Remark on Perfect Gaussian Elimination of Symmetric Matrices THOMAS ANDREAE Let M be a symmetric matrix with non-zero diagonal entries. A result of Golumbic [3, 5) states that if M has a perfect elimination scheme then it also has a perfect elimination scheme with the additional property that all pivots are chosen along the main diagonal. However, the proof given by Golumbic seems to be incomplete. In the present note, we refine Golumbic's proof, thus obtaining a complete version of it.
D. J. Rose [10] characterized the zero-non-zero schemes of those symmetric matrices for which some ordering of diagonal pivots produces no fill-in during Gaussian elimination. This result was extended by M. C. Golumbic ([3] and [5, Theorem 12.1]) who showed that Rose's characterization holds even if off-diagonal pivots are permitted. However, the proof given in [3, 5] seems to be incomplete. (This was pointed out to me by C. Scheuch.) In the present note, we refine Golumbic's proof, thus obtaining a complete version of it. For other results on the combinatorial aspects of Gaussian elimination, see also [1, 2, 4, 6, 7, 8, 9, 11] and the bibliography of [5]. For the terminology we refer to [5]. In the following, we assume familiarity with Part 1 and 2 of Golumbic [5, Chapter 12] or, likewise, with [3]. Golumbic's result can be stated as follows. THEOREM (Golumbic). Let M be a symmetric matrix with non-zero diagonal entries. If M has a perfect elimination scheme, then the associated graph G(M) is triangulated. PROOF. For shortness we write G for G(M) and B for the bipartite graph B(M) associated with M. For x E V(B), x' denotes the partner of x. The next paragraph is presented here for completeness only: except for a slight change of notation it is identical with Golumbic's argument. Suppose G has a chordless cycle VI' V2, ... , Vm, VI' m ~ 4. This corresponds in B to a graph C with V(C) = {XI, x;, ... ,xm, x~} and E(C) = {XIX;, ... ,Xm_IX~, xmx;} U {XIX~, X2X;, ... , xmx~_ d U {XIX;, ... , xmx~} (Figure 1). Let a be a perfect elimination scheme for B and let e l be the first edge of a that is incident with a vertex of C. Then el ¢ E( C) since none of the edges of C is bisimplicial in C. Put al = XI. a; = x;, and assume without loss of generality that e l = al a; for some vertex a2 ¢ C. Since in C, Adj(x l ) = {x;, x;, x~}andAdj(x;) (\ Adj(x;) (\ Adj(x~) = {xd, thebisimplicialityofe l implies that C (\ Adj(a;) = {ad and by symmetry C (\ Adj(a 2 ) = {a;}. However, this does not contradict the bisimpliciality of el when it is eliminated (as claimed in [3, 5]) but only implies that there exists an edge e2 = a2a) which was eliminated before el' We can uniquely define a subgraph L of B (Figure 1) with VeL) = V(C) U {a 2, a;, ... ,as' a;} and E(L) = E(C) U {a 2a), ... ,as_ 1 a;} u {a;a), ... ,a;_la,} u {a2 a;, ... , asa;} such that the edges ej = aja; + I are eliminated before e l (j = 2, ... , s - I) and such that as is not incident with an edge e which is eliminated before e l (s ~ 3). Let ex(I), ... , ex(s-I) be the order in which the edges e/i = 1, ... , s - I) appear in a. Note that n(s - 1) = 1. Further, let Bj be the graph that results from B by elimination of all edges of a which appear in (J before ej () = 1, ... , s - 1). Further, for all i E {I, ... , s - I}, define the graph Lj as follows: Let ejp eh, ... , ej, be the edges of
547 0195-6698/88/060547+03 $02.00/0
© 1988 Academic Press Limited
548
T. Andreae '"
,
___
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a3
s
,
-------.Osr-'
......... -
-~~--
°r,
----- . ~-e---
__ as
FIGURE
I.
a·
a·
Ji
'Ji+2
~
a·
1,'-1
a·
e·
J;+ I
'Ji-I
FIGURE
2.
{e l , ••• , es _ d which are still contained in Bj , }I <}2 < <}" and put }o = 0 and},+, = s. Then}, = I. Define L j to be the graph with V(Lj ) = V(C) u {aji: i = I, ... , t + I} u {aJi+l: i = I, ... , t} and E(L) = E(C) u {eji: i = I, ... , t} u {aj;afi-2+I: i = 2, ... , t + I} u {ajiaL,+I: i = 2, .. , t + I} (see also Fig. 2). We claim: (I)
Lj
£;
Bj (i.e. L j is a subgraph of Bj ),}
= 1, . .. , s
-
1.
Note that L = Ln(,) , and thus Bn(,) 2 Ln(l) clearly holds. For an inductive proof of (1) assume that Bn(k) 2 Ln(k) for some k(l ~ k < s - 1). We want to show that this implies Bn(k +l) 2 Ln(k+I)' Let} = n(k) and let ej"eh, ... ,ej,,}o,},+1 be as in the definition of L j . Note that this implies} = }; for a certain i ~ 2. Let B; be the graph that results from Bj by elimination of ej . Since ej = ej; is a simplicial edge of Bj , we have (see Fig. 2) aji +,aJi_2+1 E E(Bj) and, if i < t, ah +2 aL, +1 E E(B;). This implies Ln(k+l) £; B;. Hence L.(k+1) £; Bn(k+l) and thus (1) is proved. In particular, (1) implies B, 2 L and therefore (since e l is a simplicial edge of B I ) we " have: (2)
as is a neighbor of x;" ,x; and x;.
Consequently, there exists a least number k ~ 2 such that ak is a neighbor of x;", x; and It follows that a~ is a neighbor of X m , XI and Xl' Assume that k ~ 3 and note that ek_ 1 = ak_la~ E E(Bk - d. Since L k _1 £; B k _1 by (1), we find that (in B k _ l ) ak_1 is a neighbor of a vertex a; with 2 ~} ~ k - 1. Thus since ek _ 1 is a simplicial edge of Bk -I ' we find that is a neighbor of x m , XI, X2, which implies that j is a neighbor of x;", x;, x;. This contradicts the minimality of k and thus we have proved that k = 2. On the other hand, it was shown above that C n Adj(a2) = {a;}. This contradiction completes the proof. 0 X2'
a;
a
Perfect Gaussian elimination
549
REFERENCES
I. U. Derigs, O. Goecke and R. Schrader, Bisimplicial edges, Gaussian elimina tion and matchings in bipartite graphs, in Graphtheoretic Concepts in Computer Science 1984, Ed. U. Pape, pp. 79- 87, Trauner Verlag, Linz, Austria, 1984. 2. U. Faigle, O. Goecke and R. Schrader, On simplicial decomposition in combinatorial structures, Preprint Nr. 84349, Institut fUr Okonometrie und Operations Research, Universitiit Bonn, 1985. 3. M . C. Golumbic, A note on perfect Gaussian elimination, J. Math. Anal. Applics, 64 (1978), 455-457. 4. M . C. Golumbic and C . F . Goss, Perfect elimination and chordal bipartite graphs, J. Graph Theory, 2 (1978), 155-163. 5. M . C.Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York , 1980. 6. L. Haskins and D. J. Rose, Toward a characterization of perfect elimination digraphs, SIAM J. Comput., 2 (1973), 217-224. 7. D . J. Kleitman, A note on perfect elimination digraphs, SIAM J. Comput., 3 (1974), 280-282. 8. B. Korte and L. Lovasz, Greedoids, a tsructural framework for the greedy algorithm, in Progress in Combinatorial Optimization, Ed. W . R . Pulleyblank, pp. 221-243, Academic Press, New York , 1984. 9. S. L. Lauritzen, T. P . Speed and K. Vijayan, Decomposable graphs and hypergraphs, J. Austral. Math . Soc. , Ser. A,36 (1984), 12-29. 10. D . J. Rose, Triangulated graphs and the elimination process, J. Math . Anal. Applies, 32 (1970), 597-606. II. R . E. Tarjan, Graph theory and Gaussian elimination. In Sparse Matrix Computations, Ed. J. R. Bunch and D . J. Rose, pp. 3-22, Academic Press, New York, 1976.
Received 14 June 1986 THOMAS ANDREAE
II, Mathematisehes Institut der Freien Universitiit, Arnimallee 3, D-JOOO Berlin 33, F.R.G.
A Remark on Perfect Gaussian Elimination of Symmetric Matrices THOMAS ANDREAE Let M be a symmetric matrix with non-zero diagonal entries. A result of Golumbic [3, 5) states that if M has a perfect elimination scheme then it also has a perfect elimination scheme with the additional property that all pivots are chosen along the main diagonal. However, the proof given by Golumbic seems to be incomplete. In the present note, we refine Golumbic's proof, thus obtaining a complete version of it.
D. J. Rose [10] characterized the zero-non-zero schemes of those symmetric matrices for which some ordering of diagonal pivots produces no fill-in during Gaussian elimination. This result was extended by M. C. Golumbic ([3] and [5, Theorem 12.1]) who showed that Rose's characterization holds even if off-diagonal pivots are permitted. However, the proof given in [3, 5] seems to be incomplete. (This was pointed out to me by C. Scheuch.) In the present note, we refine Golumbic's proof, thus obtaining a complete version of it. For other results on the combinatorial aspects of Gaussian elimination, see also [1, 2, 4, 6, 7, 8, 9, 11] and the bibliography of [5]. For the terminology we refer to [5]. In the following, we assume familiarity with Part 1 and 2 of Golumbic [5, Chapter 12] or, likewise, with [3]. Golumbic's result can be stated as follows. THEOREM (Golumbic). Let M be a symmetric matrix with non-zero diagonal entries. If M has a perfect elimination scheme, then the associated graph G(M) is triangulated. PROOF. For shortness we write G for G(M) and B for the bipartite graph B(M) associated with M. For x E V(B), x' denotes the partner of x. The next paragraph is presented here for completeness only: except for a slight change of notation it is identical with Golumbic's argument. Suppose G has a chordless cycle VI' V2, ... , Vm, VI' m ~ 4. This corresponds in B to a graph C with V(C) = {XI, x;, ... ,xm, x~} and E(C) = {XIX;, ... ,Xm_IX~, xmx;} U {XIX~, X2X;, ... , xmx~_ d U {XIX;, ... , xmx~} (Figure 1). Let a be a perfect elimination scheme for B and let e l be the first edge of a that is incident with a vertex of C. Then el ¢ E( C) since none of the edges of C is bisimplicial in C. Put al = XI. a; = x;, and assume without loss of generality that e l = al a; for some vertex a2 ¢ C. Since in C, Adj(x l ) = {x;, x;, x~}andAdj(x;) (\ Adj(x;) (\ Adj(x~) = {xd, thebisimplicialityofe l implies that C (\ Adj(a;) = {ad and by symmetry C (\ Adj(a 2 ) = {a;}. However, this does not contradict the bisimpliciality of el when it is eliminated (as claimed in [3, 5]) but only implies that there exists an edge e2 = a2a) which was eliminated before el' We can uniquely define a subgraph L of B (Figure 1) with VeL) = V(C) U {a 2, a;, ... ,as' a;} and E(L) = E(C) U {a 2a), ... ,as_ 1 a;} u {a;a), ... ,a;_la,} u {a2 a;, ... , asa;} such that the edges ej = aja; + I are eliminated before e l (j = 2, ... , s - I) and such that as is not incident with an edge e which is eliminated before e l (s ~ 3). Let ex(I), ... , ex(s-I) be the order in which the edges e/i = 1, ... , s - I) appear in a. Note that n(s - 1) = 1. Further, let Bj be the graph that results from B by elimination of all edges of a which appear in (J before ej () = 1, ... , s - 1). Further, for all i E {I, ... , s - I}, define the graph Lj as follows: Let ejp eh, ... , ej, be the edges of
547 0195-6698/88/060547+03 $02.00/0
© 1988 Academic Press Limited
548
T. Andreae '"
,
___
'-.
a3
s
,
-------.Osr-'
......... -
-~~--
°r,
----- . ~-e---
__ as
FIGURE
I.
a·
a·
Ji
'Ji+2
~
a·
1,'-1
a·
e·
J;+ I
'Ji-I
FIGURE
2.
{e l , ••• , es _ d which are still contained in Bj , }I <}2 < <}" and put }o = 0 and},+, = s. Then}, = I. Define L j to be the graph with V(Lj ) = V(C) u {aji: i = I, ... , t + I} u {aJi+l: i = I, ... , t} and E(L) = E(C) u {eji: i = I, ... , t} u {aj;afi-2+I: i = 2, ... , t + I} u {ajiaL,+I: i = 2, .. , t + I} (see also Fig. 2). We claim: (I)
Lj
£;
Bj (i.e. L j is a subgraph of Bj ),}
= 1, . .. , s
-
1.
Note that L = Ln(,) , and thus Bn(,) 2 Ln(l) clearly holds. For an inductive proof of (1) assume that Bn(k) 2 Ln(k) for some k(l ~ k < s - 1). We want to show that this implies Bn(k +l) 2 Ln(k+I)' Let} = n(k) and let ej"eh, ... ,ej,,}o,},+1 be as in the definition of L j . Note that this implies} = }; for a certain i ~ 2. Let B; be the graph that results from Bj by elimination of ej . Since ej = ej; is a simplicial edge of Bj , we have (see Fig. 2) aji +,aJi_2+1 E E(Bj) and, if i < t, ah +2 aL, +1 E E(B;). This implies Ln(k+l) £; B;. Hence L.(k+1) £; Bn(k+l) and thus (1) is proved. In particular, (1) implies B, 2 L and therefore (since e l is a simplicial edge of B I ) we " have: (2)
as is a neighbor of x;" ,x; and x;.
Consequently, there exists a least number k ~ 2 such that ak is a neighbor of x;", x; and It follows that a~ is a neighbor of X m , XI and Xl' Assume that k ~ 3 and note that ek_ 1 = ak_la~ E E(Bk - d. Since L k _1 £; B k _1 by (1), we find that (in B k _ l ) ak_1 is a neighbor of a vertex a; with 2 ~} ~ k - 1. Thus since ek _ 1 is a simplicial edge of Bk -I ' we find that is a neighbor of x m , XI, X2, which implies that j is a neighbor of x;", x;, x;. This contradicts the minimality of k and thus we have proved that k = 2. On the other hand, it was shown above that C n Adj(a2) = {a;}. This contradiction completes the proof. 0 X2'
a;
a
Perfect Gaussian elimination
549
REFERENCES
I. U. Derigs, O. Goecke and R. Schrader, Bisimplicial edges, Gaussian elimina tion and matchings in bipartite graphs, in Graphtheoretic Concepts in Computer Science 1984, Ed. U. Pape, pp. 79- 87, Trauner Verlag, Linz, Austria, 1984. 2. U. Faigle, O. Goecke and R. Schrader, On simplicial decomposition in combinatorial structures, Preprint Nr. 84349, Institut fUr Okonometrie und Operations Research, Universitiit Bonn, 1985. 3. M . C. Golumbic, A note on perfect Gaussian elimination, J. Math. Anal. Applics, 64 (1978), 455-457. 4. M . C. Golumbic and C . F . Goss, Perfect elimination and chordal bipartite graphs, J. Graph Theory, 2 (1978), 155-163. 5. M . C.Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York , 1980. 6. L. Haskins and D. J. Rose, Toward a characterization of perfect elimination digraphs, SIAM J. Comput., 2 (1973), 217-224. 7. D . J. Kleitman, A note on perfect elimination digraphs, SIAM J. Comput., 3 (1974), 280-282. 8. B. Korte and L. Lovasz, Greedoids, a tsructural framework for the greedy algorithm, in Progress in Combinatorial Optimization, Ed. W . R . Pulleyblank, pp. 221-243, Academic Press, New York , 1984. 9. S. L. Lauritzen, T. P . Speed and K. Vijayan, Decomposable graphs and hypergraphs, J. Austral. Math . Soc. , Ser. A,36 (1984), 12-29. 10. D . J. Rose, Triangulated graphs and the elimination process, J. Math . Anal. Applies, 32 (1970), 597-606. II. R . E. Tarjan, Graph theory and Gaussian elimination. In Sparse Matrix Computations, Ed. J. R. Bunch and D . J. Rose, pp. 3-22, Academic Press, New York, 1976.
Received 14 June 1986 THOMAS ANDREAE
II, Mathematisehes Institut der Freien Universitiit, Arnimallee 3, D-JOOO Berlin 33, F.R.G.