- Email: [email protected]
A note on planar graphs
A Note on Planar Graphs by
DAVID
P.
Department University
BROWN
of Electrical of Wisconsin,
AND
ALAN
BUDNER
Engineering Madison,
Wisconsin
ABSTRACT:Some new properties of the distribution of elements and vertices with respect to the windows of a connected planar graph G are established. It is also shown that a window matrix of
G has properties similar to the properties of an incidence matrix of a
graph which is not necessarily planar. A method to form the inverse of a nonsingular submatrix of a window matrix directly from the graph is given. Introduction
Recent developments in the area of synthesis of n-port networks (1) indicate the need for further study of planar graphs (2, 3) and associated matrices. The discovery of new properties of planar graphs is important in its own right and in addition facilitates the understanding of the concept of duality and the development of synthesizing techniques. In this paper some new properties of the distribution of elements and vertices with respect to the “windows” of a connected planar graph G are established. These properties are related to a property of graphs which need not be planar and therefore give additional insight into the structure of planar graphs. It is also shown that a “window” matrix of G has properties similar to the properties of an incidence matrix of any graph. This is perhaps to be anticipated from ideas of duality. A method for constructing the inverse of a nonsingular submatrix B,, of a window matrix directly from the graph is given. In addition, it is shown that det B, can take on any positive or negative integral value. Some
Properties
of Planar
Graphs
A graph is planar if it can be mapped on a plane such that there are no crossovers, that is, no two elements of the graph have a point in common which is not a vertex. The symbol G is used in what follows to represent any connected planar graph with no noncircuit elements. Definition 1: An area of a plane bounded by the elements of any circuit of a planar graph G mapped on a plane is a region R of G. Each circuit of G defines two regions. One region, the interior region, consists of a finite area of the plane and the other region, the exterior region, consists of an infinite area of the plane. If R1 and Rz are any two regions of G, then the union of R1 and Rz, RX + Rz, is the set of points of the plane each of which is in either R1 or R2.
222
-1 hrote on Planar Graphs Dejnition 2: A window W of a planar graph G is a circuit of G the elements of which define a region R such that R is not the union of two or more regions of G. A property of the elements of a connected planar graph is given next. Theorem 1: Each element of a connected planar graph G is in two and only two windows of G. Proof: Suppose e is any element of G. Since G has no noncircuit elements, e is in some circuit C of G. The circuit C of G defines two and only two regions, RI and Rz. If RI and Rz are regions of windows, the conclusion follows. If RI, for example, is not the region of a window, it is the union of two or more (disjoint) windows and e is in some one of these windows. Since this is also true of Rz, the conclusion follows. The next theorem characterizes the element content of some windows of connected planar graphs. Theorem 2: Any connected planar graph has an even number of windows (including zero) each of which contains an odd number of elements. Proof:
If d (Wi) is the number of elements of G contained in the i-window
Wi of G, then by Theorem
1, C
d(WJ
= Ze,
WicG
where e is the number of elements in G. Since the right side of the above equation is an even number, there must be an even number (possibly zero) of odd d (WJ in C WjcGd( W;). This proves the conclusion. In any window, the number of elements is equal to the number of vertices. This fact together with Theorem 2 establishes the following theorem. Theorem 3: Any connected planar graph has an even number of windows (including zero) each of which contains an odd number of vertices. A different characterization of the vertices of connected planar graphs is given below. Theorem 4: In any connected planar graph G, there is always an even number of vertices (including zero) each of which is contained in an odd number of windows. Proof: For any graph there is always an even number of vertices of odd degree (4). The degree of a vertex of a planar graph G is equal to the number of windows containing the vertex. Therefore G contains an even number of vertices each of which is contained in an odd number of windows. The fact that a planar graph G has a dual is used to establish the number of windows in G. Theorem 5: Any connected planar graph G has e - v + 2 windows where e and v are the number of elements and vertices respectively in G.
Vol. 280, No. 3, September
1965
223
David
p.
Brown
and illan Bud??,er
proof: If Gd is a dual of G by hypothesis, then the rank of G is equal to the nullity of Gd (2). If G has e elements and v vertices, then v - 1 = e - v’ + 1 where v’is the number of vertices in Gd. Therefore v’ = e - v + 2. Since the number of vertices in Gd is equal to the number of windows in G, the conclusion follows. The Window
Matrix
of a Planar
Graph
h circuit matrix of a planar graph G (with no non-circuit elements) each row of which corresponds to a window of G is investigated in this section. The properties of this matrix are similar to those of an incidence matrix of a graph which is not necessarily planar (4). The circuit orientation of the elements in any window of a connected planar graph G is denoted below as the window orientation or simply W-orientation of the elements. Definition 3: The complete window matrix Bw = [bij] of a connected planar graph G (with no non-circuit elements) containing e elements and v vertices is of order e - v + 2 X e and
bii =
+ 1 if element j is in window i and the W- and e- orientations coincide. - 1 if element j is in window i and the W- and e-orientations do not coincide. 0 if element j is not in window i.
The next theorem, Theorem 6 follows from Theorem 1, and Theorem 7 is a consequence of Theorem 2. Theorem
6: Each
column of B,
contains two and only two non-zero
entries. Theorem 7: There is an even number of rows (including zero) of B, each of which contains an odd number of non-zero entries. The two non-zero entries in any column of B, may both be equal to +l, - 1 or one equal to + 1 and the other equal to - 1. However, the window of G can be oriented so that the non-zero entries in any column of B, are +l and -1. Theorem 8: If all the windows of G are oriented so that the region of each window is to the right (left) of the orientation arrow, that is clockwise (counterclockwise) for finite regions and counterclockwise (clockwise) for the infinite region, the non-zero entries in each column of B, are + 1 and - 1. Proof: Any element el of G is in two and only two windows WI and Wp of G. Suppose the e-orientation arrow of el coincides (does not coincide) with the WI-orientation arrow and also coincides (does not coincide) with the Wz orientation arrow. This implies that the regions of WI and Wz are both to the right (left) of el. Therefore, these regions are not disjoint. Since this contradicts the fact that WI and Wz are windows, it follows that if the e-orientation arrow of el coincides (does not coincide) with the WI-orientation arrow, it does not
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A Xote on Planar Graphs
coincide (coincides) with the Ws-orientation arrow. Therefore, the non-zero entries in the column of R, corresponding to el are +l and - 1. This completes the proof. Changing the orientation of a window of a connected planar graph G amounts to multiplying the corresponding row of B, by - 1. Therefore, if B, is any complete window matrix of G, it follows by Theorem 8, that there exists a diagonal matrix D with + 1 and - 1 diagonal entries such that the non-zero er,tries in each column of DB, are + 1 and - 1. Lemma
1: The rank of B, is at most e -
v + 1.
Proof: Let D be a diagonal matrix with + 1 and - 1 diagonal entries such that the non-zero entries in each column of DB, are +l and - 1. Since B, and DB, are of the same rank, the conclusion follows by an argument similar to that of the proof of Theorem 4-3 (4). Lemma 2: A linear combination with coefficients + 1 and - 1 of any r rows of B,, r < e - v + 2, contains at least one non-zero entry. Proof: Consider DB, as in the proof of Lemma 1. By an argument similar to that given in the proof of Lemma 4-4 (4), the sum of any r rows of DB,, r < e - v + 2, contains at least one non-zero entry. Therefore, at least one column of any set of r rows, r < e - v + 2, contains a single non-zero entry. Hence a linear combination with coefficients + 1 and - 1 of any set of r rows cannot produce a row of zeros. Theorem
9: The rank of B, is e -
v + 1.
Proof: Consider DB, as in the proof of Lemma 1. Since B, and DB, are of the same rank the conclusion follows by an argument similar to that of proof (b) of Theorem 4-4 (4). Since DB, is the complete incidence matrix of the dual of the planar graph G, Theorem 9 is also established directly from Theorem 4-4 (4). Corollary 1: If any row of B, is omitted, the resulting matrix has rank e-v+l. It is proven next that the determinant of any nonsingular submatrix of B, is fl. Theorem
10: The matrix B, is an E-matrix
(5).
Proof: Consider DB, as in the proof of Lemma 1. It follows by the argument used in the proof of Theorem 5-7 (4), that DB, is an E-matrix. Since multiplication of any row of a nonsingular submatrix by - 1 does not change the magnitude of its determinant, B, is also an E-matrix. The matrix B obtained from B, by omitting any row is a window matrix of G. The next theorem gives an interesting structural property of B. Theorem
11: A window matrix B of a connected
planar graph G can be
arranged as
Vol. 280, No. 3, September
1965
225
David P. Brown and Alan Budaer
B = [“d’
$:I,
where BK’ exists and the columns of B11 correspond to any e element subgraph of G which contains no segs and the elements of which are contained in exactly e + 1 different windows. Proof: Let G, be any e element subgraph of a connected planar graph G which contains no segs and the elements of which are contained in exactly I! + 1 different windows. Form B, for G with the elements of G, corresponding to the first e columns and the first e + 1 rows of B, corresponding to the e + 1 different windows containing the elements of G,. Omit any one of the first e + 1 rows of B, to form B. The matrix B is arranged as
where the columns of B11 correspond to the elements of G, and the rows of B11 correspond to some e windows containing the elements of G,. By this arrangeof ment Bal must be a zero matrix. Since G, contains no segs, G, is a subgraph some cotree of G (6). Therefore, the columns of B11 are linearily independent. Since Bll is square, it is nonsingular. This completes the proof. Theorem 12: If any e element subgraph G, of a connected planar graph G contains no segs and the elements of which are contained in exactly e + 1 different windows, then the elements corresponding to G, in a dual of G form a connected circuitless subgraph. Proof: By the method of forming a dual of G, the e-element subgraph corresponding to G, in the dual contains e + 1 vertices. Since G, contains no segs, the corresponding subgraph in a dual contains no circuits. It follows from properties of a tree (4) that this subgraph must be connected. This completes the proof. As an application of some of the results of this section, the rank of the complete circuit matrix of a connected planar graph is proven. Lemma 3: If B, is the complete circuit matrix of a connected planar graph, then any row of B, is a linear combination with coefficients +l and - 1 of some r rows, r < e - v + 2, of B,. Proof: Let R be any row of the complete circuit matrix B, of a connected planar graph. A region of the circuit C corresponding to R is either the region of a window or the union of r regions each of which is defined by a windovv, r < e - v + 2. If a region of C is the region of a window, then R is equal to or equal to the negative of the row of B, corresponding to this window. Therefore, suppose a region of C is the union of some r regions each of which is defined by a window Wi of G. Consider DB, as in the proof of Lemma 1 and the r rows of DB, corresponding to the Wi’s. The sum of these r rows is a row RI containing a non-zero entry in each column corresponding to an element in
226
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A Note on Pla~aar Graphs
C and a zero entry in each column corresponding to an signs of the non-zero entries in RI correspond to either wise or C being oriented counterclockwise. Therefore R Since D is diagonal with + 1 and - 1 diagonal entries, Theorem
13: The rank of B, is e -
element not in C. The C being oriented clock is equal to RI or -RI. the conclusion follows.
v + 1.
Proof: The matrix B, contains B as a submatrix except possibly for some rows multiplied by - 1. Therefore, by Lemma 3, B, is the same rank as B. Hence by Theorem 9 B, is of rank e - v + 1. A Relationship
Between
F-Circuit
and Window
Matrices
Consider the set of f-circuits Cf; for any tree of G where each C/i is oriented so that its finite region is to the right of the orientation arrow, that is clockwise. Suppose an f-circuit matrix for this set of f-circuits is B,. Let the windows It’j of G be oriented as in Theorem 8 and let I? be the window matrix obtained from some B, of G by omit,ting the row associated with the window with an infinite region. If the sequence of columns in Bf and B correspond to the same sequence of elements of G, then there is a square matrix T such that
Bf = TB, where 1) if Cl; is a window Wj, then T has a + 1 in the (i, j) position, and 2) if the finite region of Cfi is the union of the finite regions Waj: j = 1, 2, . . a, n, then T has a +l in the (i, aj) j = 1, 2, - . -, n, positions. This follows since the finite region of any circuit of a connected planar graph G, in particular, any f-circuit of G, is either the finite region of a window or the union of r finite regions each of which is defined by a window, where r
of B,
Okada (9) has proven that det B, is equal to F, where i is a non-negative integer, for certain types of circuits. It has been implied (4) that this result
Vol. 280, No. 3, September 1965
227
David
p.
Brown
and Alan Budm-
is true for all types of circuits. That this is not the case follows from an example given by Cederbaum (10) where det B, = 3. More detail is added to the properties of B, below where it is proven that det B,, for B, associated with a planar graph, can take on any positive or negative integer value (11). Before establishing the main result of this section an interesting property of determinants of matrices containing an even number of non-zero entries in each column is proven. Theorem 14: If A1 is any square matrix with entries fl, 0, containing an even number of non-zero entries in each column (or row), then det M is either zero or an even integer. Proof: If A1 has r rows, add r - 1 rows of Af to the remaining row, say the r-row. Each entry in the r-row is either zero or an even integer. This follows by hypothesis. Therefore det A1 = 3 X det Afi, where Ml is the matrix obtained by factoring 2 from the r-row of AI. Since all the entries in M are integers, det 21/1xis an integer. Thus det M is either zero or an even integer. Since the det M is equal to det M’; the conclusion follows for the case of an even number of entries in each row. The above property is useful as a rapid check on the value of the determinant of a matrix, however, the condition of the property is not a necessary condition for the determinant of a matrix to be an even integer.
FIG. 1. Theorem 15: For the planar graph of Fig. 1, having a tree in the form of a star and the circuits indicated, det B, = n where n is any positive or negative integer. Proof:
as shown.
228
Form B = [BtBc] for the graph of Fig. 1 and the circuits oriented The submatrix of B which corresponds to the cotree of Fig. 1 is
Journal
of The Franklin
Institute
A n’ote on
‘1 1 1 Oll..*ll 101.. B=110.. . . . 1
1
.
1
.
*
1
.
.
. * .
1 1 11’ .
.
.
*
0
Planar
Graphs
0’
1
where the sequence of columns is 1, 2, . . . , n + 1. Addition of - 1 times the first row to rows 3, 4, . . . , 12+ 1 results in c1 0 O-l 0
1 1
I*** 1-e. 0.e.
O-l*** .
0
10 11 01 01
0
.
.
.
o***-10
The i-row is added to the i + l-row in the sequence i = 2,3, ...,n. This results in the following triangular matrix, ‘1 1 1 Oll*.*ll 001.. 000..*13’
*
.
. . . . . OOO...On which has a determinant obtain the above matrix versing the direction of all entries in some one det B, = - n. Since n is
*
1
0.
*
1
2
.
.
equal to n. Since the operations performed on B, to do not change the value of det B,,det B, = n. Rethe orientation of any circuit in Fig. 1 results in row of B, being multiplied by - 1. For this case, any positive integer, the conclusion follows.
Corollary 2: If Gd is a dual of G of Fig. 1, then det St = n where n is any positive or negative integer and S = [S&l is a basis seg matrix of Gd with the columns of St associated with the tree of Gd corresponding to the cotree of G consisting of elements 1, 2, . . . , n + 1. Conclusion
Some new properties of planar graphs have been established. It has also been shown that a circuit matrix of a planar graph with each row associated with a window of the graph is closely related to an incidence matrix of a graph which is not necessarily planar. An interpretation, in terms of the graph, of the inverse of nonsingular submatrices of window matrices is given.
Vol. 280. No. 3, September 1965
229
David P. Brown and Alan Budner References (1) E. A. Gullemin, et al., “The Realization of n-Port Networks Without Transformers-A Panel Discussion,” IRE Trans. on Circuit Theory, Vol. CT-g, pp. 202-14, Sept., 1962. (2) H. Whitney, “Non-Separable and Planar Graphs,” Trans. Am. Math. Sot., Vol. 54, pp. 150-168, 1932. (3) H. Whitney, “Planar Graphs,” Pund. Math., Vol. 21, pp. 73-84, 1933. (4) S. Seshu and M. B. Reed, “Linear Graphs and Electrical Networks,” Addison-Wesley Pub. Co., Reading, Mass., 1961. (5) I. Cederbaum, “Matrices All of Whose Elements and Hubdeterminants are 1, - 1, or 0,” J. Afath. Phys., Vol. 36, pp. 351-361, 1958. (6) M. B. Reed, “The Seg: A Ir;ew Class of Subgraphs,” IRE Tram. on Circuit Theory, Vol. CT-S, pp. 17-22, March, 1961. (7) F. H. Branin, Jr., “The Relation Between Bron’s Method and the Classical Methods of Network Analysis,” IRE Weston Convention Record, pt. 2, pp. 3-28, 1959. (8) J. A. Resh, “The Inverse of a Nonsingular Submatrix of an Incidence Matrix,” IRE Trans. 071Circuit Theory (corresp.) Vol. CT-lo, p. 132, March 1963. (9) S. Okada, “On Node and Mesh Determinants,” Proc. IRE Corresp., Vol. 43, p. 1527, 1955. (10) I. Cederbaum, “Invariance and Mutual Relations of Network Determinants,” J. Afaath. Phys., Vol. 34, pp. 236-244, 1955. of Network Matrices,” M.8. (11) A. Budner, “Some New Properties and Interrelations Thesis, Univ. of Wisconsin, Madison, Wise., 1964.
230
Journal
of The Franklin
Institute
DAVID
P.
Department University
BROWN
of Electrical of Wisconsin,
AND
ALAN
BUDNER
Engineering Madison,
Wisconsin
ABSTRACT:Some new properties of the distribution of elements and vertices with respect to the windows of a connected planar graph G are established. It is also shown that a window matrix of
G has properties similar to the properties of an incidence matrix of a
graph which is not necessarily planar. A method to form the inverse of a nonsingular submatrix of a window matrix directly from the graph is given. Introduction
Recent developments in the area of synthesis of n-port networks (1) indicate the need for further study of planar graphs (2, 3) and associated matrices. The discovery of new properties of planar graphs is important in its own right and in addition facilitates the understanding of the concept of duality and the development of synthesizing techniques. In this paper some new properties of the distribution of elements and vertices with respect to the “windows” of a connected planar graph G are established. These properties are related to a property of graphs which need not be planar and therefore give additional insight into the structure of planar graphs. It is also shown that a “window” matrix of G has properties similar to the properties of an incidence matrix of any graph. This is perhaps to be anticipated from ideas of duality. A method for constructing the inverse of a nonsingular submatrix B,, of a window matrix directly from the graph is given. In addition, it is shown that det B, can take on any positive or negative integral value. Some
Properties
of Planar
Graphs
A graph is planar if it can be mapped on a plane such that there are no crossovers, that is, no two elements of the graph have a point in common which is not a vertex. The symbol G is used in what follows to represent any connected planar graph with no noncircuit elements. Definition 1: An area of a plane bounded by the elements of any circuit of a planar graph G mapped on a plane is a region R of G. Each circuit of G defines two regions. One region, the interior region, consists of a finite area of the plane and the other region, the exterior region, consists of an infinite area of the plane. If R1 and Rz are any two regions of G, then the union of R1 and Rz, RX + Rz, is the set of points of the plane each of which is in either R1 or R2.
222
-1 hrote on Planar Graphs Dejnition 2: A window W of a planar graph G is a circuit of G the elements of which define a region R such that R is not the union of two or more regions of G. A property of the elements of a connected planar graph is given next. Theorem 1: Each element of a connected planar graph G is in two and only two windows of G. Proof: Suppose e is any element of G. Since G has no noncircuit elements, e is in some circuit C of G. The circuit C of G defines two and only two regions, RI and Rz. If RI and Rz are regions of windows, the conclusion follows. If RI, for example, is not the region of a window, it is the union of two or more (disjoint) windows and e is in some one of these windows. Since this is also true of Rz, the conclusion follows. The next theorem characterizes the element content of some windows of connected planar graphs. Theorem 2: Any connected planar graph has an even number of windows (including zero) each of which contains an odd number of elements. Proof:
If d (Wi) is the number of elements of G contained in the i-window
Wi of G, then by Theorem
1, C
d(WJ
= Ze,
WicG
where e is the number of elements in G. Since the right side of the above equation is an even number, there must be an even number (possibly zero) of odd d (WJ in C WjcGd( W;). This proves the conclusion. In any window, the number of elements is equal to the number of vertices. This fact together with Theorem 2 establishes the following theorem. Theorem 3: Any connected planar graph has an even number of windows (including zero) each of which contains an odd number of vertices. A different characterization of the vertices of connected planar graphs is given below. Theorem 4: In any connected planar graph G, there is always an even number of vertices (including zero) each of which is contained in an odd number of windows. Proof: For any graph there is always an even number of vertices of odd degree (4). The degree of a vertex of a planar graph G is equal to the number of windows containing the vertex. Therefore G contains an even number of vertices each of which is contained in an odd number of windows. The fact that a planar graph G has a dual is used to establish the number of windows in G. Theorem 5: Any connected planar graph G has e - v + 2 windows where e and v are the number of elements and vertices respectively in G.
Vol. 280, No. 3, September
1965
223
David
p.
Brown
and illan Bud??,er
proof: If Gd is a dual of G by hypothesis, then the rank of G is equal to the nullity of Gd (2). If G has e elements and v vertices, then v - 1 = e - v’ + 1 where v’is the number of vertices in Gd. Therefore v’ = e - v + 2. Since the number of vertices in Gd is equal to the number of windows in G, the conclusion follows. The Window
Matrix
of a Planar
Graph
h circuit matrix of a planar graph G (with no non-circuit elements) each row of which corresponds to a window of G is investigated in this section. The properties of this matrix are similar to those of an incidence matrix of a graph which is not necessarily planar (4). The circuit orientation of the elements in any window of a connected planar graph G is denoted below as the window orientation or simply W-orientation of the elements. Definition 3: The complete window matrix Bw = [bij] of a connected planar graph G (with no non-circuit elements) containing e elements and v vertices is of order e - v + 2 X e and
bii =
+ 1 if element j is in window i and the W- and e- orientations coincide. - 1 if element j is in window i and the W- and e-orientations do not coincide. 0 if element j is not in window i.
The next theorem, Theorem 6 follows from Theorem 1, and Theorem 7 is a consequence of Theorem 2. Theorem
6: Each
column of B,
contains two and only two non-zero
entries. Theorem 7: There is an even number of rows (including zero) of B, each of which contains an odd number of non-zero entries. The two non-zero entries in any column of B, may both be equal to +l, - 1 or one equal to + 1 and the other equal to - 1. However, the window of G can be oriented so that the non-zero entries in any column of B, are +l and -1. Theorem 8: If all the windows of G are oriented so that the region of each window is to the right (left) of the orientation arrow, that is clockwise (counterclockwise) for finite regions and counterclockwise (clockwise) for the infinite region, the non-zero entries in each column of B, are + 1 and - 1. Proof: Any element el of G is in two and only two windows WI and Wp of G. Suppose the e-orientation arrow of el coincides (does not coincide) with the WI-orientation arrow and also coincides (does not coincide) with the Wz orientation arrow. This implies that the regions of WI and Wz are both to the right (left) of el. Therefore, these regions are not disjoint. Since this contradicts the fact that WI and Wz are windows, it follows that if the e-orientation arrow of el coincides (does not coincide) with the WI-orientation arrow, it does not
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A Xote on Planar Graphs
coincide (coincides) with the Ws-orientation arrow. Therefore, the non-zero entries in the column of R, corresponding to el are +l and - 1. This completes the proof. Changing the orientation of a window of a connected planar graph G amounts to multiplying the corresponding row of B, by - 1. Therefore, if B, is any complete window matrix of G, it follows by Theorem 8, that there exists a diagonal matrix D with + 1 and - 1 diagonal entries such that the non-zero er,tries in each column of DB, are + 1 and - 1. Lemma
1: The rank of B, is at most e -
v + 1.
Proof: Let D be a diagonal matrix with + 1 and - 1 diagonal entries such that the non-zero entries in each column of DB, are +l and - 1. Since B, and DB, are of the same rank, the conclusion follows by an argument similar to that of the proof of Theorem 4-3 (4). Lemma 2: A linear combination with coefficients + 1 and - 1 of any r rows of B,, r < e - v + 2, contains at least one non-zero entry. Proof: Consider DB, as in the proof of Lemma 1. By an argument similar to that given in the proof of Lemma 4-4 (4), the sum of any r rows of DB,, r < e - v + 2, contains at least one non-zero entry. Therefore, at least one column of any set of r rows, r < e - v + 2, contains a single non-zero entry. Hence a linear combination with coefficients + 1 and - 1 of any set of r rows cannot produce a row of zeros. Theorem
9: The rank of B, is e -
v + 1.
Proof: Consider DB, as in the proof of Lemma 1. Since B, and DB, are of the same rank the conclusion follows by an argument similar to that of proof (b) of Theorem 4-4 (4). Since DB, is the complete incidence matrix of the dual of the planar graph G, Theorem 9 is also established directly from Theorem 4-4 (4). Corollary 1: If any row of B, is omitted, the resulting matrix has rank e-v+l. It is proven next that the determinant of any nonsingular submatrix of B, is fl. Theorem
10: The matrix B, is an E-matrix
(5).
Proof: Consider DB, as in the proof of Lemma 1. It follows by the argument used in the proof of Theorem 5-7 (4), that DB, is an E-matrix. Since multiplication of any row of a nonsingular submatrix by - 1 does not change the magnitude of its determinant, B, is also an E-matrix. The matrix B obtained from B, by omitting any row is a window matrix of G. The next theorem gives an interesting structural property of B. Theorem
11: A window matrix B of a connected
planar graph G can be
arranged as
Vol. 280, No. 3, September
1965
225
David P. Brown and Alan Budaer
B = [“d’
$:I,
where BK’ exists and the columns of B11 correspond to any e element subgraph of G which contains no segs and the elements of which are contained in exactly e + 1 different windows. Proof: Let G, be any e element subgraph of a connected planar graph G which contains no segs and the elements of which are contained in exactly I! + 1 different windows. Form B, for G with the elements of G, corresponding to the first e columns and the first e + 1 rows of B, corresponding to the e + 1 different windows containing the elements of G,. Omit any one of the first e + 1 rows of B, to form B. The matrix B is arranged as
where the columns of B11 correspond to the elements of G, and the rows of B11 correspond to some e windows containing the elements of G,. By this arrangeof ment Bal must be a zero matrix. Since G, contains no segs, G, is a subgraph some cotree of G (6). Therefore, the columns of B11 are linearily independent. Since Bll is square, it is nonsingular. This completes the proof. Theorem 12: If any e element subgraph G, of a connected planar graph G contains no segs and the elements of which are contained in exactly e + 1 different windows, then the elements corresponding to G, in a dual of G form a connected circuitless subgraph. Proof: By the method of forming a dual of G, the e-element subgraph corresponding to G, in the dual contains e + 1 vertices. Since G, contains no segs, the corresponding subgraph in a dual contains no circuits. It follows from properties of a tree (4) that this subgraph must be connected. This completes the proof. As an application of some of the results of this section, the rank of the complete circuit matrix of a connected planar graph is proven. Lemma 3: If B, is the complete circuit matrix of a connected planar graph, then any row of B, is a linear combination with coefficients +l and - 1 of some r rows, r < e - v + 2, of B,. Proof: Let R be any row of the complete circuit matrix B, of a connected planar graph. A region of the circuit C corresponding to R is either the region of a window or the union of r regions each of which is defined by a windovv, r < e - v + 2. If a region of C is the region of a window, then R is equal to or equal to the negative of the row of B, corresponding to this window. Therefore, suppose a region of C is the union of some r regions each of which is defined by a window Wi of G. Consider DB, as in the proof of Lemma 1 and the r rows of DB, corresponding to the Wi’s. The sum of these r rows is a row RI containing a non-zero entry in each column corresponding to an element in
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A Note on Pla~aar Graphs
C and a zero entry in each column corresponding to an signs of the non-zero entries in RI correspond to either wise or C being oriented counterclockwise. Therefore R Since D is diagonal with + 1 and - 1 diagonal entries, Theorem
13: The rank of B, is e -
element not in C. The C being oriented clock is equal to RI or -RI. the conclusion follows.
v + 1.
Proof: The matrix B, contains B as a submatrix except possibly for some rows multiplied by - 1. Therefore, by Lemma 3, B, is the same rank as B. Hence by Theorem 9 B, is of rank e - v + 1. A Relationship
Between
F-Circuit
and Window
Matrices
Consider the set of f-circuits Cf; for any tree of G where each C/i is oriented so that its finite region is to the right of the orientation arrow, that is clockwise. Suppose an f-circuit matrix for this set of f-circuits is B,. Let the windows It’j of G be oriented as in Theorem 8 and let I? be the window matrix obtained from some B, of G by omit,ting the row associated with the window with an infinite region. If the sequence of columns in Bf and B correspond to the same sequence of elements of G, then there is a square matrix T such that
Bf = TB, where 1) if Cl; is a window Wj, then T has a + 1 in the (i, j) position, and 2) if the finite region of Cfi is the union of the finite regions Waj: j = 1, 2, . . a, n, then T has a +l in the (i, aj) j = 1, 2, - . -, n, positions. This follows since the finite region of any circuit of a connected planar graph G, in particular, any f-circuit of G, is either the finite region of a window or the union of r finite regions each of which is defined by a window, where r
of B,
Okada (9) has proven that det B, is equal to F, where i is a non-negative integer, for certain types of circuits. It has been implied (4) that this result
Vol. 280, No. 3, September 1965
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David
p.
Brown
and Alan Budm-
is true for all types of circuits. That this is not the case follows from an example given by Cederbaum (10) where det B, = 3. More detail is added to the properties of B, below where it is proven that det B,, for B, associated with a planar graph, can take on any positive or negative integer value (11). Before establishing the main result of this section an interesting property of determinants of matrices containing an even number of non-zero entries in each column is proven. Theorem 14: If A1 is any square matrix with entries fl, 0, containing an even number of non-zero entries in each column (or row), then det M is either zero or an even integer. Proof: If A1 has r rows, add r - 1 rows of Af to the remaining row, say the r-row. Each entry in the r-row is either zero or an even integer. This follows by hypothesis. Therefore det A1 = 3 X det Afi, where Ml is the matrix obtained by factoring 2 from the r-row of AI. Since all the entries in M are integers, det 21/1xis an integer. Thus det M is either zero or an even integer. Since the det M is equal to det M’; the conclusion follows for the case of an even number of entries in each row. The above property is useful as a rapid check on the value of the determinant of a matrix, however, the condition of the property is not a necessary condition for the determinant of a matrix to be an even integer.
FIG. 1. Theorem 15: For the planar graph of Fig. 1, having a tree in the form of a star and the circuits indicated, det B, = n where n is any positive or negative integer. Proof:
as shown.
228
Form B = [BtBc] for the graph of Fig. 1 and the circuits oriented The submatrix of B which corresponds to the cotree of Fig. 1 is
Journal
of The Franklin
Institute
A n’ote on
‘1 1 1 Oll..*ll 101.. B=110.. . . . 1
1
.
1
.
*
1
.
.
. * .
1 1 11’ .
.
.
*
0
Planar
Graphs
0’
1
where the sequence of columns is 1, 2, . . . , n + 1. Addition of - 1 times the first row to rows 3, 4, . . . , 12+ 1 results in c1 0 O-l 0
1 1
I*** 1-e. 0.e.
O-l*** .
0
10 11 01 01
0
.
.
.
o***-10
The i-row is added to the i + l-row in the sequence i = 2,3, ...,n. This results in the following triangular matrix, ‘1 1 1 Oll*.*ll 001.. 000..*13’
*
.
. . . . . OOO...On which has a determinant obtain the above matrix versing the direction of all entries in some one det B, = - n. Since n is
*
1
0.
*
1
2
.
.
equal to n. Since the operations performed on B, to do not change the value of det B,,det B, = n. Rethe orientation of any circuit in Fig. 1 results in row of B, being multiplied by - 1. For this case, any positive integer, the conclusion follows.
Corollary 2: If Gd is a dual of G of Fig. 1, then det St = n where n is any positive or negative integer and S = [S&l is a basis seg matrix of Gd with the columns of St associated with the tree of Gd corresponding to the cotree of G consisting of elements 1, 2, . . . , n + 1. Conclusion
Some new properties of planar graphs have been established. It has also been shown that a circuit matrix of a planar graph with each row associated with a window of the graph is closely related to an incidence matrix of a graph which is not necessarily planar. An interpretation, in terms of the graph, of the inverse of nonsingular submatrices of window matrices is given.
Vol. 280. No. 3, September 1965
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David P. Brown and Alan Budner References (1) E. A. Gullemin, et al., “The Realization of n-Port Networks Without Transformers-A Panel Discussion,” IRE Trans. on Circuit Theory, Vol. CT-g, pp. 202-14, Sept., 1962. (2) H. Whitney, “Non-Separable and Planar Graphs,” Trans. Am. Math. Sot., Vol. 54, pp. 150-168, 1932. (3) H. Whitney, “Planar Graphs,” Pund. Math., Vol. 21, pp. 73-84, 1933. (4) S. Seshu and M. B. Reed, “Linear Graphs and Electrical Networks,” Addison-Wesley Pub. Co., Reading, Mass., 1961. (5) I. Cederbaum, “Matrices All of Whose Elements and Hubdeterminants are 1, - 1, or 0,” J. Afath. Phys., Vol. 36, pp. 351-361, 1958. (6) M. B. Reed, “The Seg: A Ir;ew Class of Subgraphs,” IRE Tram. on Circuit Theory, Vol. CT-S, pp. 17-22, March, 1961. (7) F. H. Branin, Jr., “The Relation Between Bron’s Method and the Classical Methods of Network Analysis,” IRE Weston Convention Record, pt. 2, pp. 3-28, 1959. (8) J. A. Resh, “The Inverse of a Nonsingular Submatrix of an Incidence Matrix,” IRE Trans. 071Circuit Theory (corresp.) Vol. CT-lo, p. 132, March 1963. (9) S. Okada, “On Node and Mesh Determinants,” Proc. IRE Corresp., Vol. 43, p. 1527, 1955. (10) I. Cederbaum, “Invariance and Mutual Relations of Network Determinants,” J. Afaath. Phys., Vol. 34, pp. 236-244, 1955. of Network Matrices,” M.8. (11) A. Budner, “Some New Properties and Interrelations Thesis, Univ. of Wisconsin, Madison, Wise., 1964.
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