Hybrid Analysis

Book title goes here

H.Narayanan

June 11, 2009

2

Foreword

No foreword

3

Preface

No preface for the chapter.

5

6

Contents

1 Hybrid Analysis and Combinatorial Optimization 3

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Topological Hybrid Analysis Procedure . . . . . . . . . . . . . . . 81.4 Proofs for Topological Hybrid Analysis . . . . . . . . . . . . . . . 121.5 Principal Partition Problem . . . . . . . . . . . . . . . . . . . . . 131.6 Building maximally distant forests . . . . . . . . . . . . . . . . . 151.7 Network Analysis through Topological Transformation . . . . . . 20

1.7.1 Solution of the node fusion - fission problem . . . . . . . . 221.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1

2 CONTENTS

Chapter 1

Hybrid Analysis and

Combinatorial Optimization

1.1 Introduction

In this chapter we discuss the hybrid analysis problem and sketch one of its nat-ural generalizations. Focusing attention on these naturally leads to the studyof fundamental combinatorial optimization problems which can be solved us-ing the matroid union operation (see [Narayanan+Patkar09]) and the Dilworthtruncation operation (see Section 1.7).

Electrical network analysis is the process of finding pairs of vectors (v, i),such that v satisfies Kirchhoff’s Voltage Law (KVL) for the graph G of thenetwork, i satisfies Kirchhoff’s Current Law (KCL) for G and the pair (v, i)satisfies the device characteristic of the network. For ease of discussion, we willassume that the network is static (no derivative terms in the constraints) andhas a unique solution, i.e., a unique (v, i) pair satisfies the constraints of thenetwork. The basic methods of analysis reduce the constraints (KCL,KVL,devicecharacteristic) of the network to a more compact form involving node voltagesor loop currents. In the former case, once the node voltages are obtained, bythe use of KVL the branch voltage vector can be obtained uniquely and thenceusing device characteristic, the branch current vector. In the latter case, fromthe loop currents, the branch current vector can be obtained through use ofKCL and thence using device characteristic, the branch voltage vector. A naturalgeneralization of these procedures is to pick as unknowns some voltages and somecurrents from which by the use of KCL and KVL either the voltage or currentof every branch can be obtained, after which the use of device characteristic willenable us to obtain the remaining variable of the branch. This class of methodswhere the unknowns are a mixture of current and voltage variables is called‘hybrid analysis’.

Hybrid Analysis is originally due to G.Kron ([Kron39],[Kron63]) and wassimplified by Branin ([Branin62]). A description very accessible to the general

3

4CHAPTER 1. HYBRID ANALYSIS AND COMBINATORIAL OPTIMIZATION

reader is available in [Brameller+John+Scott69]. The development presentedhere is however based on a topological version reported in [Narayanan79]. It hasthe advantage of greater flexibility in the choice of unknowns and also advan-tages in storage. The manner in which voltage and current variables are chosenin hybrid analysis can be viewed in a general way as a process of transform-ing the given network through the operations of node fusions and fissions intoanother simpler network. Such a general transformation can be called ‘topolog-ical transformation of electrical networks’ [Narayanan87] and has applicationsparticularly in the parallel processing of network analysis.

Natural questions that arise are on how to choose the unknowns minimally.In the case of hybrid analysis this leads to the question of ‘Principal Partition’and in the more general case of topological transformations this leads to the‘Principal Lattice of Partitions’ problem. The former is discussed in Section 1.5and the latter is sketched in Section 1.7.

1.2 Preliminaries

We need a few preliminary definitions and results before we move on to a dis-cussion of the methods.

We assume familiarity with the notions of graph, subgraph, directed graph,path, circuit, cutset, connectedness, connected components, (spanning) tree,cotree etc. A forest of the graph is obtained by taking a tree for each componentand a coforest is its complement. For a graph G, which for us will invariablybe directed, V (G), E(G) denote the vertex set and edge set respectively. Thenumber of edges in a forest of G is its rank , denoted by r(G), and that in acoforest is its nullity, denoted by ν(G). A separator of a graph is a subset ofedges with the property that there is no circuit of the graph containing an edgeinside and an edge outside the subset. Minimal separators are called elementaryseparators . Any connected graph can be uniquely decomposed into subgraphs(called 2-connected components) on elementary separators, which will be linkedto each other at ‘hinges’ or ‘cut vertices’. A hinge is the only vertex where twoconnnected subgraphs, whose edge sets are complements of each other, meet.Electrically speaking, i.e., in terms of KCL and KVL, it is as though thesesubgraphs are disconnected.

Vectors are treated as functions from a set to a field, invariably that of realnumbers. Examples are voltage and current vectors defined on the set of edgesof a graph and potential vector defined on the set of nodes of a graph. If f is avector on S and T ⊆ S, the restriction of f to T denoted f/T, is the vector onT , whose values agree with the values of f on T . Given a collection K of vectorson S which includes the zero vector, and a subset T ⊆ S, the collection K · T ismade up of all restrictions of vectors in K and the collection K×T is the subsetof K · T where each vector is the restriction of a vector that is zero on S − T . Iff ,g are both vectors on a set S, the dot product < f ,g > ≡

∑

e∈S f(e).g(e). Thevectors f ,g are said to be orthogonal if their dot product is zero. The collectionof all vectors on S orthogonal to vectors in K, is denoted K⊥. When K is a

1.2. PRELIMINARIES 5

vector space and T ⊆ S, it can be shown directly that (K·T )⊥ = K⊥×T . Usingthe fact that when S is finite and K is a vector space, we have K⊥⊥ = K, itwould then follow that (K × T )⊥ = K⊥ · T . (See [Narayanan+Patkar09].)

The incidence matrix (usually denoted by A), of a directed graph, has onerow per node and one column per edge, with the (i, j) entry being +1(−1) if edgej is directed away (towards) node i and zero otherwise. The matrix obtainedfrom the incidence matrix, by omitting one row per component of the graph, iscalled the reduced incidence matrix and has the same row space as the incidencematrix. For a graph G, a current vector i is a vector on E(G) that is orthogonalto the rows of the incidence matrix of G, equivalently, that satisfies Kirchhoff’scurrent equations (KCE): Ax = 0 .A voltage vector v of G is a vector on E(G) that is linearly dependent on therows of the incidence matrix of G i.e., vT = λTA for some vector λ.The vector λ assigns a value to each node of G and is called a potential vector.We say v is derived from the node potential vector λ.Voltage vectors and current vectors form vector spaces denoted by Vv(G),Vi(G),and called voltage space of G and current space of G respectively. An immediateconsequence of the definition of the voltage and current spaces is the celebratedTellegen’s Theorem which states that (Vv(G))⊥ = Vi(G).

It is clear from the definition of voltage vector, that we can assign, for theedges of any tree of a connected graph, arbitrary voltage values and this woulduniquely determine cotree voltages. For, if tree voltages are given, we can assigna reference potential to some node and by traversing the tree, assign to allother nodes an appropriate unique potential. Thus the tree voltages uniquelyfix difference of potential between any pair of nodes and thence fix all cotreevoltages. In particular, we can assign to one branch e of the tree t, value 1and to all others in the tree, value 0. Let ve ≡ (ve

t |ve

t) be the corresponding

voltage vector. Note that when the branch e is removed from the tree, the lattersplits into two connected pieces with vertex sets Ve+, Ve−, say, with Ve+ beingthe vertex set where the tail of e is incident. The set of edges in the originalgraph between these two vertex sets is called the fundamental cutset of e withrespect to cotree t and denoted L∗(e, t). The vector ve has nonzero values onlyon the edges in L∗(e, t) with the value being +1(−1) if the tail is in Ve+(Ve−).The matrix, which has as rows the voltage vectors constructed in the abovemanner for each edge in a tree t, is called the fundamental cutset matrix Q

t,

with respect to the cotree t. Given any voltage vector v ≡ (vt|vt), we observe

that the vector v −∑

e∈t(v(e)ve) is a voltage vector but has zero value on alltree branches. The above traversal through tree branches shows that it must bea zero vector. It follows that the rows of the fundamental cutset matrix form abasis for Vv(G).

Now if i ≡ (it|it) is any current vector, it is clear, since ve, i are orthogonal,that < ve

t, i

t> = − < ve

t, it > = −i(e). Thus cotree current values uniquely

determine tree current values and, if all cotree current values are zero, so will alltree currents be. In particular, we can assign to one branch c of the cotree t, value1 and to all others in the cotree, value 0. Let ic ≡ (ict |i

c

t) be the corresponding

current vector. Note that when the branch c is added to the tree t, exactly


one circuit is formed, called the fundamental circuit of c with respect to thetree t denoted by L(c, t). The vector ic has non zero values only on the edgesin L(c, t), with the value being +1(−1) if the orientation of the edge in thecircuit agrees with (opposes) that of c. The matrix, which has as rows thecurrent vectors constructed in the above manner for each edge in a cotree, iscalled the fundamental circuit matrix Bt with respect to the tree t. By usingthe argument that we used for voltage vectors, it follows that the rows of thefundamental circuit matrix form a basis for Vi(G).

Let G be a graph, with E(G) ≡ E and let T ⊆ E. We now define some usefulderived graphs natural to circuit theory.

The graph Gopen(E−T ) has the same vertex set as G but with the edges of(E − T ) removed. The graph G · T is obtained from Gopen(E − T ) by removingisolated (with no edges incident) vertices.

The vertex set of Gshort (E − T ) is the set {V1, V2, . . . Vn}, where Vi is thevertex set of the ith component of GopenT , an edge e ∈ T being directed fromVi to Vj in Gshort (E − T ), if it is directed from a ∈ Vi to b ∈ Vj in G. Thegraph G × T is obtained from Gshort (E − T ) by removing isolated vertices.(The two are the same if G is connected.)

From the above construction of G · (E − T ), G × T , we have

Theorem 1.2.1 1. Maximal intersection of forest (coforest) of G with T isa forest of G · T (coforest of G × T ).

2. Union of forests of G · (E − T ) and G × T is a forest of G.

3. r(G) = r(G · (E − T )) + r(G × T ).

4. ν(G) = ν(G · (E − T )) + ν(G × T ).

Graphs obtained from G, by opening some edges and shorting others, are calledminors of G. We note that, when some edges are shorted and others open cir-cuited, the order in which these operations are performed does not affect theresulting graph. So we have,

Theorem 1.2.2 Let T1 ⊆ T2 ⊆ E. Then

1. G · T2 · T1 = G · T1

2. G × T2 × T1 = G × T1

3. G × T2 · T1 = G · (E − (T2 − T1))× T1.

We have the following important results on the voltage and current spacesassociated with minors of graphs.

Theorem 1.2.3 1. Vv(G · T ) = (Vv(G)) · T

2. Vv(G × T ) = (Vv(G)) × T

1.2. PRELIMINARIES 7

Proof : i. Let vT ∈ Vv(G · T ). Now Vv(G · T ) = Vv(Gopen(E − T )).Thus, vT ∈ Vv(Gopen(E−T )). Let vT be derived from the potential vector λ ofGopen(E − T ). Now for any edge e ∈ T , vT (e) = λ(a)− λ(b), where a, b are thepositive and negative end points of e. However, λ is also a potential vector of G.Let the voltage vector v of G be derived from λ. For the edge e ∈ T , we have,as before, v(e) = λ(a)− λ(b). Thus, vT = v/T and therefore, vT ∈ (Vv(G)) · T.Hence Vv(G · T ) ⊆ (Vv(G)) · T . The reverse containment is proved similarly.

ii. Let vT ∈ Vv(G × T ). We have Vv(G × T ) = Vv(Gshort (E − T )).The vertex set of Gshort (E−T ) is, say, the set {V1, V2, . . . Vn}, where Vi is thevertex set of the ith component of GopenT , an edge e ∈ T being directed fromVi to Vj in Gshort (E − T ), if it is directed from a ∈ Vi to b ∈ Vj in G.

Now, vT ∈ Vv(Gshort (E−T )). Let vT be derived from the potential vector

λ in Gshort (E − T ). The vector λ assigns to each of the Vi, the value λ(Vi).

Define a potential vector λ on the nodes of G as follows: λ(n) ≡ λ(Vi), n ∈ Vi.Since {V1, . . . Vk} is a partition of V (G), it is clear that λ is well defined. Let v

be the voltage vector derived from λ in G. Whenever e ∈ E − T we must havev(e) = 0 since both end points must belong to the same Vi.Next, whenever e ∈ T we have v(e) = λ(a) − λ(b) where a is the positive endpoint of e and b, the negative endpoint. Let a ∈ Va, b ∈ Vb,where Va, Vb ∈V (Gshort (E − T )). Then the positive endpoint of e in Gshort (E − T ) is Va

and the negative end point, Vb. By definition λ(a)− λ(b) = λ(Va)− λ(Vb).Thusv/T = vT . Hence, vT ∈ (Vv(G))× T . Thus, Vv(G × T ) ⊆ (Vv(G)) × T .The reverse containment is proved similarly, but using the idea, that if a voltagevector is zero on all elements of E − T , then a potential vector from which itis derived, must have the same value on all vertices of each Vi, since these arevertex sets of components of GopenT .Using duality we can now prove

Theorem 1.2.4 Let G be a directed graph on edge set E. Let T ⊆ E. Then,

1. Vi(G . T ) = (Vi(G))× T.

2. Vi(G × T ) = (Vi(G)) · T.

Proof :

i. Vi(G . T ) = (Vv(G . T ))⊥ by Tellegen’s Theorem. By Theorem 1.2.3,Vv(G . T ) = (Vv(G)) · T. Hence, Vi(G . T ) = ((Vv(G)) · T )⊥ = (Vv(G))⊥ × T =Vi(G)× T.

ii. The proof is similar.

It is useful to note some elementary facts about coloops and selfloops. Acoloop, by definition, does not belong to any circuit and therefore must belongto every forest. Dually, a selfloop, by definition, does not belong to any cutsetand therefore must belong to every coforest. The fundamental circuit and cutsetmatrices that result when coloop (selfloop) edges are shorted or open circuitedare the same. So we have,


Theorem 1.2.5 Let T ⊆ E be a set of edges composed entirely of self loopsand coloops. Then Vv(G · (E − T )) = (Vv(G × (E − T )) and Vi(G · (E − T )) =(Vi(G × (E − T )).

We are now in a position to state and prove a result which will enable us togive a topological version of hybrid analysis.

Theorem 1.2.6 Let (A, B) be a partition of E(G). Let K be a forest and LA,a coforest of G ·A and tB be a forest and L, a coforest of G ×B. Let GAL be thegraph G × (A ∪ L) and let GBK be the graph G · (B ∪K). Then

1. iK |iLA|itB|iL is a current vector of G, iff there exist current vectors

iK |iLA|iL of GAL and i′K |itB

|iL of GBK .

2. vK |vLA|vtB|vL is a voltage vector of G, iff there exist voltage vectors

vK |vLA|v′L of GAL and vK |vtB

|vL of GBK .

For ease of readability we relegate the proof of this result to Section 1.4.

1.3 Topological Hybrid Analysis Procedure

In this section we present a topological version of the hybrid analysis procedure.Essentially, the network is decomposed into two subnetworks, whose simultane-ous analysis, matching certain ‘boundary conditions’, is equivalent to the anal-ysis of the original network. The validity of this procedure rests on Theorem1.2.6 and requires only that the device characteristic of the subsets of edges ofthe derived networks appear decoupled in the original network. In the specialcase of resistive networks we could write nodal equations for one of the networksand loop equations for the other and match boundary conditions. This yieldshybrid analysis equations with greater freedom in the choice of unknowns whichcould translate to good properties such as sparsity for the coefficient matrix.

Let N be a network on the graph G. (The reader may use Figure 1.1 toillustrate the procedure.) Let (A, B) be a partition of E(G) such that in thedevice characteristic of N the devices in A, B are independent of, i.e., decoupledfrom, each other. (In Figure 1.1, A ≡ {1, 2, 3, 4, 5} and B, the complement.) LetK be a forest and LA, a coforest of G ·A and tB be a forest and L, a coforest ofG × B. (K ≡ {1, 2, 5}, LA ≡ {3, 4}, tB ≡ {6, 8, 9}, L ≡ {7, 10, 11}). Let GAL bethe graph G × (A ∪ L) and let GBK be the graph G · (B ∪K). (In the presentexample, G × (A ∪ L) would have 7, 10 as self loops. In the figure, the graphshown with caption NAL is this graph with the self loops omitted. The graphG · (B ∪ K) would have 1, 2 as coloops. In the figure, the graph shown withcaption NBK is this graph with the coloops omitted.)

We now build two networks NAL and NBK as follows:- NAL has graph GAL

with edge set A ∪ L built from G by short circuiting (fusing the end points of)edges in tB and removing them. The devices in A have the same characteristicsas in N and L has no device characteristic constraints. NBK has graph GBK

with edge set B ∪K built from G by open circuiting edges (removing the edges

1.3. TOPOLOGICAL HYBRID ANALYSIS PROCEDURE 9

1 2

3

5

6 7

8

109

1 2

3

45

11i11

v5

116 7

8

9 10

4

N

NAL

NBK

Figure 1.1: To illustrate the NAL −NBK Method

but leaving the end points in place) in LA. The devices in B have the samecharacteristics as in N and K has no device characteristic constraints. (Notethat the L, K edges are present in both networks.) Theorem 1.2.6 implies that[Narayanan79] solving N is equivalent to solving NAL and NBK simultaneouslykeeping iL, vK the same in both networks. It may be noted that, if branchesin L contain selfloops in the graph GAL, they may be deleted or contracted(endpoints fused and edge removed) from that graph and the current matchingbetween NAL and NBK be confined to the remaining L branches. Similarly,if branches in K contain coloops in the graph GBK , they may be deleted orcontracted from that graph and the voltage matching between NAL and NBK

be confined to the remaining K branches.Consider now the case where the device characteristic for the edges has the

form(i− J ) = G(v − E), (1.1)

where G is a block diagonal matrix with principal diagonal submatrices GA, GB ,where we take GB to be invertible. In this case, hybrid analysis equations canbe written as follows:-

1. Write nodal analysis equations for NAL treating branches in L as currentsources of value iL.

2. Write loop analysis equations for NBK treating branches in K as voltagesources of value vK .


3. Force the constraints that iL is the same in both networks and vK is thesame in both networks.

We now proceed formally. The reader is invited to refer to Figure 1.1.Let [ArAArL] = [ArKAr(A−K)ArL] be a reduced incidence matrix of GAL.

Let the device characteristic of the edges in A be expressible as

(iA − JA) = GA(vA − EA). (1.2)

We then have, (since i ∈ Vi(G)),

ArAiA + ArLiL = 0 (1.3)

i.e., ArA(iA − JA) + ArLiL = −ArAJA (1.4)

i.e., ArAGA(vA − EA) + ArLiL = −ArAJA (1.5)

i.e., ArAGAvA + ArLiL = −ArAJA + ArAGAEA (1.6)

Now,[

vA

vL

]

=

[

ATrA

ATrL

]

vnA, (1.7)

for some vnA (since

[

vA

vL

]

∈ Vv(GAL)). We thus have,

(ArAGAATrA)vnA + ArLiL = −ArAJA + ArAGAEA. (1.8)

These are the nodal analysis equations of NAL. Note that we could have usedany matrix which has as its rows a basis of Vv(GAL), in place of (ArA|ArL),resulting in a valid set of equations with ‘voltage type’ of unknowns.

Next for NBK , we choose the forest t for building the fundamental circuitmatrix of GBK . Let[

BKBB

]

≡[

BKBt∩BBL

]

≡[

BKBt∩BIL

]

be the fundamental circuit matrix of GBK with respect to forest t (where IL

denotes the identity matrix with its columns corresponding to L).Let the device characteristic in B be expressible as vB − EB = RB(iB − JB).We then have,

BKvK + BBvB = 0 (1.9)

i.e., BKvK + BB(vB − EB) = −BBEB (1.10)

i.e., BKvK + BBRB(iB − JB) = −BBEB (1.11)

i.e., BKvK + BBRBiB = −BBEB + BBRBJB . (1.12)

We have,[

iBiK

]

=

[

BTB

BTK

]

y (1.13)

1.3. TOPOLOGICAL HYBRID ANALYSIS PROCEDURE 11

for some y, since

[

iBiK

]

∈ Vi(GBK). Hence,

BKvK + BBRBBTBy = −BBEB + BBRBJB. (1.14)

Here again we could have used any matrix which has as its rows a basis ofVi(GBK), in place of (BB|BK), resulting in a valid set of equations with ‘currenttype’ of unknowns.

Now we impose the condition that vK is the same in both networks and sois iL. But this means,

ATrKvnA = vK (1.15)

and,ITL y = iL (1.16)

So we get the hybrid equations,

ArAGAATrAvnA + ArLiL = −ArAJA + ArAGAEA (1.17)

BKATrKvnA + BBRBBT

BiL = −BBEB + BBRBJB (1.18)

The matrix

[

ArAGAATrA ArL

BKATrK BBRBBT

B

]

is positive definite if GA, RB are pos-

itive definite. This matrix will usually not be very sparse unless GBK has asuitable basis for its current space which makes BBRBBT

B sparse. In prac-tice this may often be possible. The real power of these methods, however,is revealed when we try to use iterative methods. We show in Section 1.4 thatBKAT

rK = −ATrL. This fact is computationally useful. Indeed, this means that

we can use a variation of the conjugate gradient method to solve Equations 1.17and 1.18 [Narayanan07][Siva09]. The advantage of such methods is that thematrix need not be stored explicitly - storing GAL, GBK and the device charac-teristic is adequate. The basic subroutine for the conjugate gradient method onlyrequires multiplication of the coefficient matrix by a given vector which arisesat each iteration. This process can be carried out entirely by breaking it downinto graph theoretic operations and multiplication by the device characteristicmatrix (which would often be nearly diagonal).

We note that if in GAL, there are selfloops in L and in GBK there are coloopsin K, then in Equations 1.17 and 1.18, the matrix entries corresponding respec-tively to the current and voltage variables associated with such branches wouldbe zero. We would therefore be justified in open circuiting or short circuitingsuch edges before we write equations.

These methods were originally derived for parallelization of network analysisby G.Kron and were called ‘Diakoptics’ [Kron63]. They exploit the fact thatwhen GAL, GBK have several 2-connected components, the matrix

[

ArAGAATrA ArL

BKATrK BBRBBT

B

]

will have block diagonal structure within ArAGAATrA and BBRBBT

B.


1.4 Proofs for Topological Hybrid Analysis

We need a preliminary lemma for the proof of Theorem 1.2.6.

Lemma 1.4.1 Let A, B, L, K be defined as in Theorem 1.2.6. Then,

1. Vi(GAL ·A) = Vi(G · A) and dually, Vv(GAL ·A) = Vv(G ·A).Vi(GBK ×B) = Vi(G ×B) and dually Vv(GBK ×B) = Vv(G ×B).

2. r(GAL) = r(G · A); ν(GBK) = ν(G ×B).

Proof of the Lemma: It can be seen that K ∪ tB is a forest and LA ∪ L, acoforest of G. It follows that K ∪ tB is a forest of G · (A ∪ tB) and LA ∪ L is acoforest of G×(B∪LA). Since K∪tB is a forest of G·(A∪tB) and K is a forest ofG·(A∪tB)·A(= G·A), it follows that (using Theorem 1.2.1) r(G·(A∪tB)×tB) =r(G · (A∪ tB))− r(G ·A) = |tB|. So ν(G · (A∪ tB)× tB) = 0. Thus the edges of tBare not part of any circuit in G · (A∪ tB). Similarly, since LA ∪L is a coforest ofG × (B ∪LA) and L is a coforest of G × (B ∪LA)×B(= G ×B), it follows that(using Theorem 1.2.1) ν(G×(B∪LA)·LA) = ν(G×(B∪LA))−ν(G×B) = |LA|.So r(G × (B ∪ LA) · LA) = 0. Thus the edges of LA are not part of any cutsetin G × (B ∪ LA).

We observe that GAL ·A ≡ G×(A∪L) ·A = G ·(A∪tB)×A. But in the graphG ·(A∪tB ), tB is a set of coloops. Hence shorting or opening these edges will notaffect the KCL or KVL constraints of the resulting graph. Thus, Vi(GAL ·A) =Vi(G × (A∪L) ·A) = Vi(G · (A ∪ tB)×A) = Vi(G · (A∪ tB) ·A) = Vi(G ·A) anddually, Vv(GAL · A) = Vv(G ·A).

Further, we note that GAL is obtained by shorting the branches tB in theforest K ∪ tB of G. Hence K, which is a forest of G · A is also a forest of GAL.This proves that r(GAL) = r(G ·A).

Next, GBK × B ≡ G · (B ∪ K) × B = G × (B ∪ LA) · B. But in the graphG× (B∪LA), LA is a set of selfloops. Hence shorting or opening these edges willnot affect the KCL or KVL constraints of the resulting graph. Thus, Vi(GBK ×B) = Vi(G·(B∪K)×B) = Vi(G×(B∪LA)·B) = Vi(G×(B∪LA)×B) = Vi(G×B)and dually, Vv(GBK ×B) = Vv(G ×B).

Further, we note that GBK is obtained by opening the branches LA in thecoforest L ∪LA of G. Hence L, which is a coforest of G ×B is also a coforest ofGBK . This proves that ν(GBK) = ν(G ×B).

Proof of Theorem 1.2.6:

1. iK |iLA|itB|iL is a current vector of G, iff there exist current vectors

iK |iLA|iL of GAL and i′K |itB

|iL of GBK .

2. vK |vLA|vtB|vL is a voltage vector of G, iff there exist voltage vectors

vK |vLA|v′L of GAL and vK |vtB

|vL of GBK .

1.5. PRINCIPAL PARTITION PROBLEM 13

Let iK |iLA|itB|iL be a current vector of G. From Theorem 1.2.3, it follows that

iK |iLA|iL is a current vector of GAL and itB

|iL is a current vector of G × B.But Vi(GBK ×B) = (Vi(GBK)) ·B = Vi(G ×B). So there exists a current vectori′K |itB

|iL of GBK . Next, suppose there exist current vectors iK |iLA|iL of GAL

and i′K |itB|iL of GBK . Since i′K |itB

|iL is a current vector of GBK , it follows thatitB|iL is a current vector of GBK ×B and therefore of G ×B. Now L ∪ LA is a

coforest of G. Hence for any arbitrary vector iLA|iL, there exists a unique current

vector i”K |iLA|i”tB

|iL of G. But then i”K |iLA|iL is a current vector of GAL and

i”tB|iL is a current vector of G × B, i.e., of GBK × B. Now L is a coforest of

G ×B and L∪LA is a coforest of GAL. So for any arbirary vector iLA|iL, there

is a unique current vector i3K |iLA|iL of GAL and i3tB

|iL of G ×B. But we alreadyhave seen that there exist current vectors iK |iLA

|iL of GAL and itB|iL of G ×B.

It follows that iK = i”K and itB= i”tB

and therefore iK |iLA|itB|iL is a current

vector of G. The voltage part of the theorem is dual to the above, i.e., in theproof above we interchange voltage and current, GAL and GBK

′·′ and ′×′, Kand L, tB and LA.Proof of claim BKAT

rK = −ATrL.

Observe that the rows of (BK |BtB|IL) span the space Vi(GBK). Hence the rows

of (BK |IL) span the space (Vi(GBK))·(K∪L) = Vi(GBK×(K∪L)), by Theorem1.2.3. But GBK = G · (B ∪ K). Hence the rows of (BK |IL) span the spaceVi(G · (B ∪K)× (K ∪ L)) = Vi(G × (A ∪ L) · (K ∪ L)). Next, we note that therows of (ArK |ArL) span the space (Vv(GAL)) · (K ∪ L) = Vv(GAL · (K ∪ L)) =Vv(G×(A∪L)·(K∪L)). The claim follows since rows of (BK |IL) and (ArK |ArL)are orthogonal.

1.5 Principal Partition Problem

In this section we relate the hybrid analysis problem to the principal parti-tion of graphs. The latter has played a fundamental role in the development ofcombinatorial optimization in the context of matroids [Narayanan+Patkar09]and submodular functions [Narayanan97]. All the results in the present sectionare best understood in a unified way as applications of the matroid union the-orem and the principal partition for polymatroid rank functions discussed in[Narayanan+Patkar09]. However, for readability we give a self contained, graphbased, treatment here while pointing out connections to results of that chapter.

When the network N is linear, if we write nodal equations for NAL and loopequations for NBK (defined as in Subsection 1.3), the total number of equationswould be r(G . A)+ ν(G ×B). So one could ask for the partition A, B for whichthe above expression reaches a minimum value. More generally, given a partition(A, B) of E(G), by Theorem 1.2.3 we know that there exists a current vectoriA|iB of G iff iB is a current vector of G × B and that there exists a voltagevector vA|vB of G iff vA is a voltage vector of G · A. Thus vA,iB of G can beuniquely determined using only KVL and KCL from the forest voltage vectorvtA of G ·A and the coforest current vector iLB

of G ×B. Thus we have the firstformulation of the hybrid rank problem:


Given a graph G, partition E(G) into A and B such that r(G . A)+ν(G×B)is minimized.

Historically, the following problems, related to the first formulation, weresolved at about the same time. After stating them we give their solution inbrief. Detailed solution may be found in the references cited therein as well asin [Narayanan97].i. The topological degree of freedom of an electrical network

[Ohtsuki+Ishizaki+Watanabe70] This problem was posed by G.Kron.Select a minimum sized set of branch voltages and branch currents from which,by using Kirchchoff’s voltage equations and Kirchhoff’s current equations, wecan find either the voltage or the current associated with each branch. This min-imum size is called the topological degree of freedom of the network, equivalently,the hybrid rank of the graph.ii. The Shannon switching game [Edmonds65b]G is a graph with one of its edges say eM ‘marked’. There are two players - a ‘cut’player and a ‘short’ player. The cut player, during his turn, deletes (opens) anedge leaving the end points in place. The short player, during his turn, contractsan edge, i.e., fuses its end points and removes it. Neither player is allowed totouch eM . The cut player wins if all the paths between the end points of eM

are destroyed (equivalently, all circuits containing eM are destroyed). The shortplayer wins if the end points of eM get fused (equivalently, all cutsets containingeM are destroyed by shorting of edges). The problem is to analyze this gameand characterize situations where the cut or short player, playing second, canalways win and to determine the winning strategy.iii. Maximum distance between two forests [Kishi+Kajitani69]Define distance between two forests t1 and t2 as |t1 − t2|. Find two forests in agiven graph which have the maximum distance between them, i.e., the size oftheir union is the largest possible.iv. Forest of minimum size hybrid representation [Kishi+Kajitani69]Let a forest t be represented by a pair of sets (At, Bt) where At ⊆ t,t ∩ Bt = ∅such that (At1 , Bt1) = (At2 , Bt2) iff t1 = t2. Note that we can represent thesame forest by several pairs, for instance (t, ∅), (∅, E(G) − t) both represent t.We call |At

⋃

Bt| the size of the representation (At, Bt). Find a forest which hasthe representation of minimum size.v. The maximum rank of a cobase submatrix [Iri69]For a rectangular (m× n) matrix with linearly independent rows, let us call anm× (n−m) submatrix a cobase submatrix iff the remaining set of columns arefrom an identity matrix. The term rank of a matrix is the maximum number ofnonzero entries in the matrix which belong to distinct rows and distinct columns.Find a cobase matrix of maximum rank, and a cobase matrix of minimum termrank among all matrices row equivalent to the given matrix.

For the above five problems the solution involves essentially the same strat-egy: Find a set A (or a minimal set Amin or a maximal set Amax) whichminimizes 2r(G · A) + |E(G) − A|. That these sets are unique is proved inTheorems 1.6.1 and 1.6.2. The partition of the graph into Amin, Amax −Amin, E(G)−Amax was called the Principal Partition of G by Kishi and Kajitani

1.6. BUILDING MAXIMALLY DISTANT FORESTS 15

[Kishi+Kajitani69]. This has been discussed in detail for the more general caseof polymatroid rank functions in [Narayanan+Patkar09]. The essential ideaswill be repeated for graphs in Section 1.6 of this chapter. Below we have givena sketch of the solutions to the five problems. More details may be found in[Narayanan97].

Let tA be a forest of the subgraph on A. Let LA be a coforest of the graphon G × (E(G)−A). Select the branch voltages of tA and the branch currents ofLA as the desired set of variables.

If eM ∈ Amin, the short player can always win. If eM ∈ (E(G) − Amax) thecut player can always win. If eM ∈ Amax−Amin, whoever plays first can alwayswin. The winning strategies involve the construction of appropriate maximallydistant forests during every turn.

Kishi and Kajitani gave an algorithm for building a pair of maximally distantforests which is essentially the well known algorithm for building a base of theunion of two matroids (see [Edmonds65a] for the case where the matroids areidentical - essentially the same algorithm works for the general case).

Select a forest t which has maximal intersection with A. The representation(t∩A, (E(G)−t)

⋂

(E(G)−A)) has the least size among all representations of allforests. As is easily seen, the minimum size among all representations of forestsof G is also the same as the topological degree of freedom of G and the abovemaximum distance.

The solution is similar for the last problem. Let S be the set of all columnsand let r(·) be the rank function on the collection of subsets of S. Then the max-imum rank of a cobase matrix = the minimum term rank of a cobase matrix.Select two maximally distant bases (bases ≡ maximally independent columns).In this case the matroid union algorithm described in [Narayanan+Patkar09], es-sentially a generalization of the algorithm for building maximally distant forests,has to be used. Perform row operations so that an identity matrix appears core-sponding to one of these. The submatrix corresponding to the complement ofthis base is the desired cobase matrix which has both maximum rank as well asminimum term rank.

1.6 Building maximally distant forests

The five problems stated in the previous section were solved originally withoutreference to the matroid union theorem. Indeed, the problems of finding max-imally distant forests, of minimal representation of forests and the topologicaldegree of freedom were solved graph theoretically. The algorithm for buildingmaximally distant forests of a graph is essentially the same as the above men-tioned matroid union algorithm except that both the matroids whose union issought are the polygon matroids of the same graph. We sketch this algorithminformally in this section and also relate it to the principal partition of a graphas described by Kishi and Kajitani [Kishi+Kajitani69]. We hope that this treat-ment helps in better visualization of the more general matroid ideas.

We begin with a simple but useful observation in the following lemma.


Lemma 1.6.1 Let G be a graph. Let t1, t2 be two forests and let t1, t2 be thecorresponding coforests of G. Then the following are equivalent

1. t1, t2 are maximally distant;

2. |t1 ∪ t2| is the maximum possible;

3. |t1 ∩ t2| is the minimum possible;

4. t1, t2 are maximally distant.

Proof : Since the sizes of all forests are the same, maximizing |t1 − t2| is thesame as maximizing |t1 ∪ t2| and minimizing |t1 ∩ t2|. Sizes of all coforests arethe same. So these hold also for coforests. But maximizing |t1 ∪ t2| is the sameas minimizing |t1 ∩ t2|. So t1, t2 being maximally distant is the same as t1, t2being maximally distant.

Algorithm Maximally Distant Forests

Let t1, t2 be two forests of G = (V, E). Define a directed graph G(t1, t2) with Eas the set of vertices and with directed edges as described below.Whenever in G, e /∈ tj , j = 1, 2, draw, in G(t1, t2), directed edges from the vertexe to all vertices which are edges of G in the fundamental circuit L(e, tj) (formedwhen e of G is added to tree tj) and mark each directed edge as a tj edge.From a given vertex es in G(t1, t2), it is easy to determine the set of all verticesthat can be reached through directed paths by using breadth first search. In theprocess, the shortest path (in terms of number of edges in the path), from es toevery vertex in G(t1, t2), can be determined.

The present algorithm starts from some pair of forests t1, t2 (which couldeven be the same forest) and tries to build another pair t′1, t

′2 for which the

distance |t′1 − t′2| is greater than the distance |t1 − t2|. Clearly, if t1 ∪ t2 coversall edges in E, the forests are maximally distant. Suppose eout /∈ t1 ∪ t2. InG(t1, t2), we find the set of all vertices (which are edges of G) reachable fromvertex eout through directed paths. If no vertex in this set corresponds to anedge common to both t1 and t2, we repeat the process with another such eout

from which an edge ecom common to both t1 and t2 may be reached. If no suchvertex eout exists we stop and output the current t1, t2 as maximally distant.

Let eout, (t1), e1, (t2), · · · , ei, (tj), ei+1, · · · , ek, (tj), ek+1, · · · , en−1, (tm), ecom

be a shortest path from eout to ecom, where er, (tp), er+1 indicates that from thevertex er there is a directed edge to er+1, which is marked tp, where tp couldbe either t1 or t2 (the indices used for trees and edges being unrelated). We willuse this path to alter t1, t2. The key idea we use is the following: L(ei, tj) doesnot have ek+1 as a member as otherwise we could have shortened the path toeout, (t1), e1, (t2), · · · , ei, (tj), ek+1, · · · , en−1, (tm), ecom. Therefore in the foresttj = tj∪ek−ek+1, the fundamental circuit L(ei, tj) will be the same as L(ei, tj).

We can therefore alter t1 and t2 as


tm ← tm ∪ en−1 − ecom

· · ·tj ← tj ∪ ei − ei+1

· · ·t1 ← t1 ∪ eout − e1

The result of the above alteration would be that eout would now move into t1∪t2while ecom /∈ t1 ∩ t2. Therefore the size |t1 ∪ t2| and the distance |t1 − t2| wouldhave increased.

We repeat the above step until from none of the eout we can reach any ecom

and output the current t1, t2 as maximally distant.By Lemma 1.6.1, equivalently, t1, t2 may be output as maximally distant.More directly, the above algorithm can be converted into one which finds

maximally distant coforests by working with coforest ti in place of forest ti andreplacing L(ei, tj) wherever it occurs, by the fundamental cutset L∗(ei, tj).We know that ek ∈ L∗(ei, tj) iff ei ∈ L(ek, tj). This yields the relationshipbetween G(t1, t2) and G(t1, t2) stated in the following lemma.

Lemma 1.6.2 Reversing the direction of arrows in G(t1, t2) and marking tjedges by tj yields G(t1, t2).

We justify the above algorithm through the following theorem. Herer(A), ν(A) denote respectively r(G ·A), ν(G ×A).

Theorem 1.6.1 Let t1, t2 denote forests of G on edge set E and let A ⊆ E.Then,

1. |t1 ∪ t2| ≤ 2r(A) + |E − A| and the inequality becomes an equality onlyif t1, t2 are maximally distant and ti ∩ A, i = 1, 2, are disjoint forests ofG . A and (t1 ∪ t2) ∩ (E −A) = (E −A);

2. the pair of forests t1, t2 output by Algorithm Maximally Distant Forests andthe set of all edges Amin in E corresponding to vertices in G(t1, t2) whichcan be reached from E− (t1 ∪ t2) satisfy |t1 ∪ t2| = 2r(Amin)+ |E−Amin|and hence t1, t2 are maximally distant;

3. max |t1 ∪ t2| = min 2r(A) + |E − A|, where t1, t2 are forests of G(V, E)and A ⊆ E;

4. if t1, t2 denote coforests of G, then |t1 ∪ t2| ≤ 2ν(A) + |E − A| and theinequality becomes an equality iff t1, t2 are maximally distant, ti ∩ A, i =1, 2, are disjoint coforests of G ×A and (t1 ∪ t2) ∩ (E −A) = (E −A).

Proof : i. We have ti∩A, i = 1, 2, as a subforest of G . A and (t1∪t2)∩(E−A) ⊆(E−A). So |t1 ∪ t2| ≤ 2r(A)+ |E−A|. It is clear that if the inequality becomesan equality, the corresponding t1, t2 must be such that |t1 ∪ t2| is a maximumand therefore be maximally distant. The inequality becomes an equality iff the


set A on the RHS is such that |(t1 ∪ t2) ∩ A| = 2r(A), i.e., ti ∩ A, i = 1, 2, aredisjoint forests of G . A and (t1 ∪ t2) ∩ (E −A) = (E −A).

ii. If the t1, t2 output by the algorithm are such that t1 ∪ t2 = E, theinequality will be satisfied as an equality by taking Amin to be the null set. Ifon the other hand, the algorithm outputs t1, t2 such that t1∪ t2 6= E, then Amin

has the following properties. First, it contains E − (t1 ∪ t2), or equivalently,(t1 ∪ t2) ⊇ (E − Amin) and Amin does not contain any edge in t1 ∩ t2. Nextin G . Amin, every edge in ti ∩ Amin, i = 1, 2, can be reached from an edgeoutside t1 ∪ t2 by repeatedly taking fundamental circuits with respect to thetwo forests. Thus ti ∩ Amin, i = 1, 2, span both each other as well as edges inE − (t1 ∪ t2). So ti ∩ Amin, i = 1, 2, are both forests of G . Amin and, further,have no intersection since Amin does not contain any edge in t1 ∩ t2. Thus wehave, |t1 ∪ t2| = 2r(Amin) + |E −Amin|.

iii. is now immediate from i. and ii.iv. The proof for the coforest case is dual, i.e., by replacing in the above

argument, forests by coforests, G(t1, t2) by G(t1, t2), Amin by A∗min, G . Amin

by G ×A∗min and r(G . Amin) by ν(G ×A∗

min).

Theorem 1.6.2 Let t1, t2 (t1, t2) be maximally distant forests (coforests) ofgraph G on edge set E. Let Amin (Bmin) denote the set of all edges in E corre-sponding to vertices in G(t1, t2) (G(t1, t2)) which can be reached from E−(t1∪t2)(E − (t1 ∪ t2)).

1. Amin(Bmin) is the unique minimal set that minimizes 2r(A)+ | E − A |(2ν(A∗)+ | E − A∗ |).

2. A ⊆ E minimizes 2r(A)+ | E−A | iff E−A minimizes 2ν(A∗)+ | E−A∗ |.

3. E −Bmin (E −Amin) is the unique maximal set that minimizes 2r(A)+ |E −A | (2ν(A∗)+ | E −A∗ |).

4. An edge e belongs to Amin (e belongs to Bmin) iff there exist maximallydistant forests t1, t2 (coforests t1, t2) s.t. e ∈ (E− (t1∪ t2)), (e ∈ (E− (t1∪t2))),

Proof : i. We know by Theorem 1.6.1, that a subset A minimizes 2r(A)+ |E − A |, A ⊆ E iff for every pair of maximally distant forests t1, t2 |t1 ∪ t2| =2r(A) + |E − A| and that this happens iff ti ∩ A, i = 1, 2, are disjoint forests ofG . A and (t1∪t2)∩(E−A) = (E−A), i.e., A ⊇ E−(t1∪t2). Thus in G(t1, t2), wesee that A contains all vertices corresponding to E−(t1∪t2) and further it is notpossible to reach outside A from within since each ti ∩ A, i = 1, 2, is a forest ofG . A. Thus Amin ⊆ A. However Amin itself minimizes 2r(A)+ | E−A |, A ⊆ Eand so is the unique minimal minimizing set.The proof for the dual statement follows by arguing with coforests t1, t2 andG(t1, t2).ii. We have, t∩A is a forest of G . A iff t∩ (E −A) is a coforest of G × (E −A)(Theorem 1.2.1 of Preliminaries). Next, ti∩A, i = 1, 2, are disjoint iff (E−A) ⊇


t1 ∩ t2, i.e., iff (E − A) ⊇ (E − (t1 ∪ t2)) and ti ∩ (E −A), i = 1, 2, are disjointiff A ⊇ t1 ∩ t2,i.e., iff A ⊇ E − (t1 ∪ t2). From Theorem 1.6.2 we know that theexpression 2r(A)+ | E −A | reaches a minimum iff ti ∩A, i = 1, 2, are disjointforests of G . A and A ⊇ E − (t1 ∪ t2) and the expression 2ν(A∗)+ | E − A∗ |reaches a minimum iff ti∩ (E−A), i = 1, 2, are disjoint coforests of G× (E−A)and (E −A) ⊇ (E − (t1 ∪ t2)). The result follows.

iii. This follows immediately from i. and ii. above.iv. Observe that every e ∈ (t1 ∪ t2) ∩ Amin can be reached from someeout ∈ Amin − (t1 ∪ t2). In G(t1, t2) we therefore have a shortest path from eout

to e, say eout, (t1), e1, (t2), · · · , ei, (tj), ei+1, · · · , en−1, (tm), e. Modifying t1, t2 asin Algorithm Maximally Distant Forests would give us a new pair of maximallydistant forests t′1, t

′2 which would not contain e as a member. The result for

Bmin follows by using G(t1, t2).

N2

b

a

N1

N3

N4

16

11

16

Figure 1.2: Network N to Illustrate the Fusion-Fission Method


N1

N2

iv=0vi=0

i

b

a1

v

a2

N3

N4

11

16

1

6

Figure 1.3: A Network equivalent to N with Virtual Sources

1.7 Network Analysis through Topological Trans-

formation

We say that we use topological transformation while analysing networks, if atintermediate stages of the analysis, we modify the topology of the network. In-stances of such transformations are the construction of two derived networksduring topological hybrid analysis, multiport decomposition etc. In this sec-tion we consider a fairly general class of transformations which we may callthe fusion-fission method, and also sketch certain optimization problems whicharise naturally during this study and which generalize the principal partitionproblem. Detailed description of these ideas may be found in [Narayanan90],[Narayanan91] and [Narayanan97].

Fusion-Fission method:

Consider the network in Figure 1.2. Four subnetworks have been connectedtogether to make up the network. Assume that the devices in the subnetworksare decoupled. Clearly the networks in Figure 1.2 and Figure 1.3 are equivalent,provided the current through the additional unknown voltage source and thevoltage across the additional unknown current source are set equal to zero. Butthe network in Figure 1.3 is equivalent to that in Figure 1.4 under the additionalconditions

iv1 + iv2 + i = 0

vi3 + vi4 − v = 0.

1.7. NETWORK ANALYSIS THROUGH TOPOLOGICAL TRANSFORMATION 21

V V

N1N2

i

i

N3

N4

i v1 i v2

vi3

vi4

11

16

Figure 1.4: Network N decomposed by the Fusion-Fission Method

(The variables iv1, iv2, vi3, vi4 can be expressed in terms of currents and voltagesof the original graph so that the additional constraints will involve only oldcurrent and voltage variables and the new variables i, v.)

As can be seen, the subnetworks of Figure 1.4 are decoupled except for thecommon variables v and i and the additional conditions.A natural optimization problem here is the following:

Given a partition of the edges of a graph into E1, · · ·Ek, what isthe minimum size set of node pair fusions and node fissions by whichall circuits (equivalently cutsets) passing through more than one Ei

are destroyed?

In the present example the optimal set of operations is to fuse nodes a and b andcut node a into a1, a2 as in Figure 1.3. Artificial voltage sources are introducedacross the node pairs to be fused and artificial current sources are introducedbetween two halves of a split node.

The above formulation can be stated in a more convenient form as

Node fusion - fission problemLet G be a graph and let Πs be a specified partition of E(G) s.t. G . Ni isconnected for each Ni ∈ Πs. Find a minimum length sequence of node pairfusions and node fissions which, when performed on G, result in a graph Gnew

in which each circuit intersects only one of the blocks of Πs (equivalently eachcutset intersects only one of the blocks of Πs).

Assuming each G . Ni to be connected is reasonable electrically speakingsince, if they are not, we can usually be working with their connected compo-nents rather than with themselves while analysing.


We will show that this problem generalizes the hybrid rank problem (seeSection 1.5). We first note that the result of a node fission followed by a nodefusion can always be achieved by a node fusion followed by a node fission. Thuswhenever we have a sequence of node fissions and fusions needed for convertinga graph into another, the result can always be achieved by a sequence of nodefusions followed by a sequence of node fissions.

We think of the original network as being made up of a number of singleedge networks and the problem is to decouple them in the above manner. At theend of the fusions and fissions no edges belonging to different networks shouldbelong to the same circuit. Therefore every edge should have become a selfloopor a coloop. Let A, B be the subsets of edges which are selfloops and coloopsrespectively. Every edge in A has its endpoints fused. This is exactly equivalentto fusing the endpoints of the edges of a forest of G ·A. Thus with r(G ·A) nodefusions all edges in A would have become selfloops. After these node fusions theresulting graph would be G × B. In any graph, making all edges into coloopscan be achieved with minimum number of operations by cutting each coforestedge at one of its end points. Therefore making all edges in B into coloopsrequires ν(G×B) node fissions. Thus the minimizing of r(G ·A)+ ν(G×B) overall partitions A, B is equivalent to finding a minimum length sequence of nodefusions and fissions which will make every edge in the graph into a selfloop orcoloop.

1.7.1 Solution of the node fusion - fission problem

In this subsection we give a sketch of the ideas involved in the solution of thenode fusion-fission problem. Let G be a graph and Πs, a partition of E(G). Thefusion rank of G relative to Πs is the minimum length of a sequence of node pairfusions needed to destroy every circuit that intersects more than one block ofΠs. The fission rank of G relative to Πs is the minimum length of a sequenceof node fissions needed to destroy every circuit that intersects more than oneblock of Πs. The hybrid rank of G relative to Πs is the minimum length of asequence of node pair fusions and node fissions needed to destroy every circuitthat intersects more than one block of Πs.

Now consider the situation where we use both fusions and fissions, with allthe fusions occurring first. Any sequence of node pair fusions would ultimatelyfuse certain groups of nodes into single nodes. Hence, as far as the effect ofthese node pair fusions on the graph is concerned, we may identify them with apartition of V (G) (with singleton blocks being permitted) each block of whichwould be reduced to a single node by the fusions. The number of node pairfusions required to convert a set of nodes V to a single node is (| V | −1).Hence, if Π is a partition of V (G), the number of node pair fusions required togo from G to the graph obtained from G by fusing blocks of Π into single vertices,which we shall denote by Gfus·Π, is | V (G) | − | Π | . This number we wouldhenceforth call, the fusion number of Π. The fission rank of Gfus·Π relative toa partition Πs of E(G) would be called the fission number of Π relative to Πs.

1.7. NETWORK ANALYSIS THROUGH TOPOLOGICAL TRANSFORMATION 23

The sum of the fusion number and the fission number of Π relative Πs wouldbe called the fusion - fission number of Π relative to Πs. Our task is to find apartition of V (G) which minimizes this number.

We now define a bipartite graph which relates Πs to V (G). Let BG be thebipartite graph associated with G, with ‘left vertices’ VL ≡ V (G) and ‘rightvertices’ VR ≡ E(G), with e ∈ VR adjacent to v ∈ V iff edge e is incident on vin G. Let B(Πs) be the bipartite graph obtained from BG by fusing the rightvertices in the blocks of Πs and replacing parallel edges by single edges.Let X ⊆ V (G). Let | ΓL | (X) denote the size of the set of right vertices adjacentto vertices in X , (| ΓL | −λ)(X) denote | ΓL | (X)− λ. Let Π be a partition ofV (G). We define (| ΓL | −λ)(Π) to be the sum of the values of (| ΓL | −λ) onthe blocks of Π. We then have the following result whose proof we omit in theinterest of brevity.

Theorem 1.7.1 Let G be a connected graph. Let Πs be a partition of E(G) s.t.G . Ni is connected for each Ni ∈ Πs. Let Π be a partition of V (G). Then

1. the fusion - fission number of Π relative to Πs equals

(| ΓL | −2)(Π)+ | V (G) | − | Πs | +1

2. the hybrid rank of G relative to Πs equals

min((| ΓL | −2)(Π)+ | V (G) | − | Πs | +1),

Π a partition of V (G).

The problem of minimizing (| ΓL | −2)(Π) over partitions of V (G) falls un-der computing ‘Dilworth truncation’ of a submodular function. The PrincipalLattice of partitions problem is that of computing all partitions which minimize(| ΓL | −λ)(·) for some λ. Strongly polynomial algorithms are available for thispurpose. Details are available in [Narayanan90], [Narayanan91], [Narayanan97].

There are strong analogies between the principal partition (minimize r(X)−λ|X |, r(·) submodular, over subsets of a given set) and the principal lattice ofpartitions (minimize (r − λ)(·), r(·) submodular, over partitions of a given set)problems. In the case of principal partition, if λ1 > λ2, a minimizing set corre-sponding to the former is always a subset of any minimizing set correspondingto the latter. In the case of principal lattice of partitions, if λ1 > λ2, a minimiz-ing partition corresponding to the former is always finer than any minimizingpartition corresponding to the latter [Narayanan97]. In addition, in the case ofgraphs, the principal partition problem of Kishi-Kajitani (on the edge set of agraph) can be posed, as described earlier, as that of finding an optimal nodefusion- fission problem and therefore can actually be solved as a principal latticeof partitions problem on the vertex set of the graph [Narayanan97]. Indeed thecurrent fastest principal partition algorithm for graphs is actually of this type[Patkar+Narayanan91].

24 BIBLIOGRAPHY

1.8 Notes

The hybrid analysis notion is peculiar to network theory giving rise nat-urally to the hybrid rank problem. This problem and its generalizationscan be regarded as unifiers for large parts of combinatorial optimization in-cluding the theory of submodular functions. This theme has been enlargedin [Narayanan97]. The principal partition and principal lattice of partitionshave many practical applications particularly in building partitioners for largescale systems (see [Patkar+Narayanan09]) and in structural solvability of sys-tems ([Ozawa76],[Ozawa+Kajitani79], see [Narayanan+Patkar09] for more ref-erences).

Bibliography

[Brameller+John+Scott69] A. Brameller,M. John and M. Scott: Practical Di-akoptics for Electrical Networks (Chapman and Hall, London, England,1969).

[Branin62] F.H.Branin Jr.: The relationship between Kron’s method and theclassical methods of network analysis. Matrix and Tensor Quarterly 12

(1962) 69-105.

[Edmonds65a] J. Edmonds: Minimum partition of a matroid into independentsubsets. Journal of Research of the National Bureau of Standards 69B

(1965) 67-72.

[Edmonds65b] J. Edmonds: Lehman’s switching game and a theorem of Tutteand Nash-Williams. ibid. 69B (1965) 73-77.

[Iri69] M. Iri: The maximum rank minimum term rank theorem for the pivotaltransformations of a matrix. Linear Algebra and Its Applications 2 (1969)427-446.

[Kishi+Kajitani69] G. Kishi and Y. Kajitani: Maximally Distant Trees andPrincipal Partition of a Linear Graph. IEEE Transactions on Circuit The-ory CT-16 (1969) 323-329.

[Kron39] G. Kron: Tensor Analysis of Networks (J.Wiley, New York, 1939).

[Kron63] G. Kron: Diakoptics - Piecewise Solution of Large Scale Systems (Mc-Donald,London,1963).

[Narayanan79] H. Narayanan: A theorem on graphs and its application to net-work analysis. Proceedings of IEEE International Symposium on Circuitsand Systems (1979) 1008-1011.

[Narayanan87] H. Narayanan: Topological transformations of electrical net-works. International Journal of Circuit Theory and its Applications 15

(1987) 211-233.

BIBLIOGRAPHY 25

[Narayanan90] H. Narayanan: On the minimum hybrid rank of a graph relativeto a partition of its edges and its application to electrical network analysis.International Journal of Circuit Theory and its Applications 18 (1990) 269-288.

[Narayanan91] H. Narayanan: The principal lattice of partitions of a submodu-lar function. Linear Algebra and its Applications 144 (1991) 179-216.

[Narayanan97] H. Narayanan: Submodular Functions and Electri-cal Networks. Annals of Discrete Mathematics 54 North Hol-land (London, New York, Amsterdam) (1997). (Revised version athttp://www.ee.iitb.ac.in/ hn/book/).

[Narayanan07] H. Narayanan: Mathematical Programming and Electrical Net-work Analysis II: Computational Linear Algebra through Network analy-sis, International Symposium on Mathematical Programming for DecisionMaking: Theory and Applications (ISMPDM07), ISI Delhi, January 10-11,2007.

[Narayanan+Patkar09] H.Narayanan and S.B.Patkar: Matroids, chapter in thepresent book.

[Ohtsuki+Ishizaki+Watanabe70] T.Ohtsuki, Y.Ishizaki and H.Watanabe:Topological Degrees of Freedom and Mixed Analysis of ElectricalNetworks. IEEE Transactions on Circuit Theory CT-17 (1970) 491-499.

[Ozawa76] T. Ozawa: Topological conditions for the solvability of active linearnetworks. International Journal of Circuit Theory and its Applications 4

(1976) 125-136.

[Ozawa+Kajitani79] T. Ozawa and Y.Kajitani: Diagnosability of Linear ActiveNetworks. IEEE Transactions on Circuits and Systems CAS-26 (1979)485-489.

[Patkar+Narayanan91] S. Patkar and H. Narayanan: Fast algorithm for theprincipal partition of a graph. Proceedings of Eleventh Annual Symposiumon Foundations of Software Technology and Theoretical Computer Science(FSTTCS-11) LNCS-560 (1991) 288-306.

[Patkar+Narayanan09] S.B.Patkar and H.Narayanan: Partitioning, chapter inthe present book.

[Siva09] V. Siva Sankar, H. Narayanan, and Sachin B. Patkar: Exploiting HybridAnalysis in Solving Electrical Networks, 22nd International Conference onVLSI Design,New Delhi, January 5-9, 2009, 206-210.

Index

cobase submatrix, 14coforest, 4coloop, 7current vector, 5

dot product, 4

forest, 4fundamental circuit, 6fundamental circuit matrix, 6fundamental cutset, 5fundamental cutset matrix, 5fusion-fission method, 20

graph, 4hybrid rank, 22

hinge, 4hybrid rank problem, 13hybrid representation, 14

incidence matrix, 5

matrixterm rank, 14

maximally distant forestsalgorithm for, 16

maximally distant trees, 14

nullity, 4

orthogonal vectors, 4

partitionfission number, 22fusion - fission number, 23fusion number, 22

potential vector, 5

principal lattice of partitions, 4, 23principal partition, 4, 13

rank, 4

self loop, 7separator, 4

elementary, 4Shannon switching game, 14

Tellegen’s Theorem, 5topological degree of freedom, 14topological hybrid analysis, 8topological transformation, 20

voltage vector, 5

26

Documents

Hybrid Analysis