Research Report 277
Hybrid Bases in Graphs
Ladislav A Novak and Alan Gibbons
RR2II
ln this paper we introduce a new concept, that of hybrid base, which is a maximal circuitless and
cutsetless subset of a graph. Although this concept of simultaneous circuitlessness and
cutsetlessness has been used in proofs of some theorems in so called hybrid gaph theory, it has
not received much attention. Only largest circuitless and cusetless subsets (hybrid bases of
maximum cardinality) have been recognised as important and then only as an auxiliary notion. [n
contrast to nu.x-i*rally circuitless subsets (rees) or to maximally cutsetless subsets (conees),
hybrid bases are not of the same cardinality. This fact, although seemingly an "imperfection", is
the cause of rich stnrcture which we describe in this paper through several propositions. The
concept of hybrid bases is related to several important notions in hybrid orientated gaph theory.
For example, it is related to maximally distant pairs of trees, to complementary pairs of trees, to
perfect trees and to topological degree of freedom. It is also closely related to the problem of
finding the minimum numhr of independent variables in the hybrid analysis of electrical
networks.
Dclparlment of Cornputer Science
University of Wa: wick
Coventry CV4 7AIUnited Kingdom
March 1992
Hybrid Bases in Graphs
Ladislav A Novak
Deparrnent of Elecrical Engineering, Univenity of Novi Sad, Yugoslavia
and
Alan Gibbons *
Deparunent of Computer Scierrce, University of Warwick, England
Abstract
In this paper we introduce a new concept, that of hybrid base, which is a maximal circuitless and cutsetless subset of
a graph. Although this concept
of simultaneous circuitlessness and cutsetlessness has been used in proofs of
some
theorems in so called hybrid graph theory, it has not received much attention. Only largest circuitless and cutsetles
subsets (hybrid bases
of maximum cardinality) have been recognised
notion. In contrast to maximally circuitless subsets (fees) or
o
as important and then only as an auxiliary
maximally cutsetless subsets (cotrees), hybrid
bases
are not of the same cardinality. This fact, although seemingly an "imperfection", is the cause of rich structure which
we describe in this paper through several propositions. The concept of hybrid bases is related
notions
in hybrid orientated graph
theory. For example,
complementary pairs of trees, to perfect pairs of trees and
it is related !o maximally
o topological
tJo
several important
distant pairs of trees, !o
degree of freedom. It is also closely related
o
the problem of finding the minimum number of independent variables in the hybrid analysis of electrical retworks.
$1 Introduction
We intnrduce a new concept, that of hybrid base, which we believe has relevance in current graph
theory and within certain important applications.
of a graph is said to be a hybrid base of the graph if it is both circuitless and
cutsetless and maximal in the sense that no other circuitless and cutsetless subsets of the graph
contain b as a proper subset. For exarnple, figure I shows two copies of the same graph with trvo
A
subset b of edges
different hybrid bases indicated with bold lines. Any subset of a hybrid base is also both circuitless
and cutsetless and will be called a double independent subset of the graph. It is obvious that a
hybrid base or any of its subsets may be seen as part of a tree and part of a cotree at the same time,
and consequently has rhe properties of both. But there is one important distinction: in contrast to
maximally circuitless subsets (trees) o{'the gaph or to maximally cutsetless subsets (corees) of tne
graph, maximally lcircuitless and cutsetless subsets (hybrid bases) arc not necessarily of the same
cardinality. This is the case, for example, for the graph of figure 1. This "irnperfection" however
turns out to be an advantage.
x Partially sr.rpported by the ESPRIT II BRA Prognimme of the EC under critltract No. 3075 (ALCOIO.
bold cdges bclorg
b hybrid bases
lbrl=
3
l\tl=
2
(b)
(a)
Figue I
Although the concept of simultaneous circuitlessness and cutsctlessness has been used in prmfs of
some theorems U,2,3\, it is surprising that this concept has not received much anention. Only
largest cfucuitless and cutsetless subses have been recognised as important [3,4] but not rrcre than
as an
auxiliary notion.
The part of graph theory which deals with concepts which are inherently related to both circuits and
cutsets, we shall call hybrid gmph theory. The promotion of new notions can be a qpurious process
and
it may not at first
be clear which are
of value and whether the most useful defrnitions have been
made. We hope that the notion of hybrid base
and
will take its place as a notion of independent
will provide excellent intuitive insight within the area of hybrid
value,
graph theory.Throughout this
paper we shall, without loss of generality, be concemed only with 2-connected graphs.
It was pointed out by Amari [5] that the set of edges of a graph can be divided into nvo distina
subsets such that the sum of the rank of one subset and the corank of the other may give a number
which is less than both the rank and the corank of the gpph. Not long afterwards, this number was
forrnally defined by Tsuchiy a et al. and called ttr hybrid rank t6J. In the sam paps the notion of a
minimal hybid rank taken over all possible par:titions of the edge set of a gnph was inrodrrced and
called the tqlogical degree of fieedom. That paper together with tbc paper of Kishi and Kajitani []
in which maximally distant pain of trees and principal partition werc introduced providc the
foundation of the hybrid approach in graph theory. Since 1967 many Papers have been published
in this area mostly by
Japanese authors.
The concept of hybrid bases is *learly related to several important notions in hybrid orientated
gaph theory. For example, it i5 r;;".lated to maximally distant pain of trees [], to complementary
pairs of l,'ees [3], ro Fedscl pairs of trees [8] and to topological degree of freedom [6,7] . It is also
related to the problem of frnding the minimum number of independent variables in the hybrid
analysis of elecrical networks.
S2
Preliminaries
'lJris sei:tion is devoted to some definitions a"nd assertions related to material that follows. Wc
prcslrme that the reader is familiar v,'ith the following basic notions of graph theory: graph, edge,
circuit arrd cutset We take these to L* prirna"ry notions drat need not tre defined. However we will
defrne all other notions on the basis of these. Throughout we denote a graph by G and im edge set
3
by E. The tenns circuit, cutsct, trec, cotree, forest and coforest will bc uscd here o rncan a subcct
of edges of a gnph. A forest is a maximal circuitless subset of edges while a coforest is a maximal
cutsetless subset of edges.
If the graph
is connected then a forest is a tnee and a coforest is a corec.
by t*. The non-negative integer ranl F,
related to a tree t, is called the diameter of the tree t. Given a tree t, any edgc in the corrcsponding
In what follows, a tree wiX be denoted by t and
a cotree
cotree t* forms exactly one circuit with edges in t" Such a circuit is called a fundanrental circuit
of G with resp€ct to t. Similarly, any edge of the tnee t defines exactly one cutset with the edges in
the corresponding cotree t*. Swh a cutset is called a fundamentel cutset of G with reqp€ct to
t
.
If E' is a subset of E then the rank of E' is the cardinality of the largest circuitless srbsct of E, tbc
co-rank of E' is the cardinality of the largest cutsetless subset of E and the complement of E'is
the set difference E\E' denoted by E*. By lE'l we denote the number of elemens in (that is, thc
cardinality o0 the subset of E'. Given a subsets s of a graph, we denote by nr(s) (rcspectively
nr(s)) the maximum number of independent cutsets (circuits) made of edges in s only. The
distance between two (spanning) trees tl arrd b of a graph, written ltr\|, is the number of
edges which are in tt but not in t . A tree g is said to be maximally distant from another trec tl
if ltr\t l ) ltr\l for every trer t of the graph. A pair of trees (tr,b) is defined to be a perfect pair of
rees if both t2 is maximally distant from tt and tt is maximally disant fro- b. A pair of trees (tr,g)
is defined to be a maximally distant pair of trees if ltt\l > lt\"| for every pair (t',t'). A pair of
rees (t1,9) is said to be a complementary pair of trees if tt and harc disjoint and their unim
covers the edge set E.
Assertion
I
t8l
Given a tree to of a graph G, (Vt)
W
<rank to*
Assertion 2 [8]
re equivalenc
distant from tt
The following five statements
i)
t2 is maximally
ii) the fundamental circuit with respectto t2defined by an edge in tt*nb* contains no edges in
tt^b.
iii) the fundamental cutset with respect to t2* defined by an edge in ttnb contains no edges in
ti*nt2*.
iv) ltr\gl- rank tr*
v) ltrngl
= ns(tl)
Assertion
3
tEl
The following five statements are aluivalent:
i)
(tr,tz) is a perfect pair
ii)
fundamental circuits with respect to tt and t2defined by edges in
trnh'
tt*nt r'contains no edges in
I
iii) fundamental cutsets with respcct !o tlf and t r defined
by dgqs in
trnb
contains no edges in
tt*nb*.
iv) rank tt* =
v)
nr(t1) =
ltt\l =lh\tl=
rank t2*
hrnbl= nr(!)
In proving theorems we shall occasionally refer to the following three theorcms which one
found in the graph theoretic literaure, frexample in [0,11].
can
Orthogonality theorem
Given a graph G let C be a circuit and S bc a cutset of G. Then
Painting theorem (also known
as the
Cn Sl* l.
Colouring theorem)
Given a gaph G let {e}, Et and Q form a partition of the edge set E of G. Then either e forms
circuit with edges in E, only
s
a cutset
wi&
edges in
Q
a
only, but not bodr.
Maximal independence theorem
Irt
A
be any edge subset of a graph G.
Th€n all maximal circuitless (cutsetless) subses of A
re of
the same carrdinality.
A more general version of the Painting theorem, together with a large number of its coollaries
be found in [12] .
can
$3 Double independence and hybrid base
A
subset of edges of a graph G is said to bc a double independent subset
of E if it contains no
circuits and no cutsets of the gaph G. This concept, although it has never been given a specific
name has been used in proofs of some thorems f1,2,3,41.
A double independent
subset d of
*dges
of a graph G is said to be a hybrid base of G if it is
maximal in the sense that no other double independent sul)set of C contain d as a ploper suset [n
other words a double independent subset d of G is a hybrid base iff for an arbitrary edge e in d*,
dw{e} is not a double
independent subset of G. Obviously, an edge subset of G is a double
independent subset of G iff it is a subset of a hybrid base of G.
In contrast to maxirnally circuitless subsets {tiees) or to maxin'rally r;ulselless subsetr
r;i;rximally circuitless and cutsetl*ss subsets (hybrid 505,',s)
illustrations we ples€nt the following exanrples:
neerJ ncit
(cotrees),
have the same cardinaliry. As
5
Example
I
For the graph shown in figure 2 all hybrid bases are listed bclow and classified by cardinaliry into
two groups
a)
with cardinality 2:
(3,6)
b) with cardinality 3:
(1,3,4)
(1,5, O
(2,4,6)
(1, 3, 5)
(1,4,
q
(2, 3,
(2,3,
o
t
(2,5,6)
6
Figure 2
This is one
d
the smallest examples with at least two different cardinalities of hybrid bascs
boll
eilges
bebng O
bytriil
[\r=
?
lbel=
6
baser
hl= 5
Figure 3
Example 2
We employ an augmented version of the g'aph of figure 2, shown in figure 3. All hybrid bases arc
classified by cardinality in three gruups: with cardinality 5, with cardinality 6 and with cardinaliry
7. One representative for each group is thown in figure 3. The list of all hyb'rid bases for tlre g.aph
is shown in Appendix 1.
Remark I
Starting with
the graph of exarnple
l, it is possible to build
a class of graphs by consecutively
addiirg new 'floors' which are copies of the same glaph . Adding one floor above the ground floor
gives the graph of figure 3 for which
thrrc different cardinalities of hybrid bases: 5,6 and
7 . The result of building n- I floors over the ground floor, is sho.,vn in figure 4.
"chcre arr:
w
!r
tl
rl
lb{=3n+D-1
ta)
ll
I /I\
vl)
i--l-
D+l
tl
w
Figure 4
It is not difficult to prove thu all hybrid bases of the graph obtained (which has n floors including
the ground floor) can be classified by cardinality into n+l groups represented by carrdinalities 3n-1,
3n , ..., 4n-1.
We now present some properties of double independent subsets and hybrid bases based on the
Painting theorem. The first proposition gives a necessary and sufficient condition for a subset of
edges to be a double independent subset
Proposition
A subset d of
circuit ard
I
edges is a double independent subset
a cutset
of
a graph
iff
each edgc
in d forms both
a
with edges in d* only.
Proof
Irt an edge e belongs to a subset d- Then according to Painting theorem
applied on triple ( d\[e],
{e}, d*) e does not form either a circuit or a cutset with edges in d only iff e forms both
and a cutset with edges in d*
only.
lrtdbea
1.
2
double independent subset of a graph and let eed*. Ttren
independcnt
iff
circuit
tr
Next proposition is a strengtren venion of Proposition
Proposition
a
e belongs to a
du [e] is also double
circuit and to a cutset made of edges in d* only.
Proof
riple (4 [e],
d\[e]),
both a circuit
and a cutset nr.ide of edges in d* only iffe does not form either a circuit or a cutset with edgcs in d
only. Becausc d is double independent, the last staiement means that d t-r [e] is also double
o
independent
Ac,conding to the Painting theorem applied to the
e belongs to
7
Note
In proving Proposition l, Painting tlreorem is apptied to the riple ( dr{e}, {e}, d*), whereas in the
case of Proposition 2 this theorem is applied to the triple (d, {e}, d\{c}). The difference is in the
choice of the edge c.
Now we establish necesary and sufficient conditions for a double independent subset to be
a
hybrid base.
Proposition 3
Given a double independcnt subset d of a graph the following statements are equivalent
r) d is a hybrid basc
ii) any circuit and any cutset madc of edges in d* only are disjcint
iii)
any edge in d* forms acircuit and/or a cuts€t with edges in d only
Proof
i)
<=+
ii) A double irdependent subset d is a hybrid base if for each €€ d*, dt-r{e} is not double
independenl But according O Proposition 2 il is true iffthere is no edge that belongs at dle same
time to a circuit and a cutset made of edges in d* only, that is, iff any circuit and any cutset made of
edges in
d* only, are disjoinr
ii)<+iii) The statement ii) is tnre iff there is no edge in d* that fmms both a circuit and a cutset with
edgesin d* only, thatis, iffforeachedge eed+, edoesnot formacircuit and/oracutsetwith
edges in d*
only. By applyrng
the Painting
e forms a cutset and/or a circuit with edges
The following algorithm for finding
Algorithm
a
theorem to the triple (d, {e},
in d
d\[e]),
we deduce that
only.
tr
hybrid base is based on hoposition 3.
I
(to find a hybrid base b)
Input: A gnph G.
begin
f +- se t r:f all edges of the graPh G
b<-6
while f is nonempty do
begin
Contract the edges in G that belong to b and in the graph obtained identify edges that belong to
self circuits. Denote the set of these edges by C.
Rernove the edges in G that belong to b and in the graph obtained identify edges that
belong to self cutsets. Denote the set of these edges by S.
f <.- the complement of the union b u C t-r S
Choose any edge eef
b <-- bu{e}
end
end of Algorithm
t
According to Prroposition 3, each edge in the complement of a hybrid basc b, forms a circuit or a
curser with edges in b only. Because a hybrid base is always part of a ftc, a circuit that an edge in
the complernent of b forms with edges in b only is a fundamental circuit with rcspcct o the uec and
consequently is unique. From dual reasoning, a cutset that an edge in the complement forms with
in b only is also unique. But, nevertheless, an edge from the complement of b may form u
the same time both a circuit and a cutset. For example, in the gaph of figure 5, the marked edges
which belong to the corylement of a hybrid base (forrned frm ttre bold edges) -ake both a circuit
edges
and a cutset with the hybrid base.
Figure 5
Thus, the edges in d* can be classified into three groups: edges that do not form cutscts with edges
in d only, edges that do not form circuits with edges in d only and edges that form both a circuit
and a cutset wirh edges in d only. According to the Painting theorem an edge belongs to tbe furst
iff it belongs
to a circuit (a cutset) made of edgcs in dr only.
According to the same theorem an edge belongs to the third group iff it does not belong to either a
circuit or a cuts€t made of edges in d* only.
(respectively, second) gfoup
g4 Rank considerations
Recall n, and n.. Given a double independent subset d of a gaph G, the fcrllowing proposition
relates the numbers
and
d*
nr(d*) and n"(d*) to the carrdinaliry of d and dt to the rank and tre corank of d
and to ttre rank and the corank of G.
Proposition
4
Given a graph G and a hybrid base d of G, tire following relations hold:
rankG = ldl+nr(d*) =rankd* = ld*l- n.(d*), rankd = ldl
corankG= Hl + n"(d*)=cordnkdr =ld*l-nr(dt), corunkd =ldl
Proof
This follows the statt:iixnt and proof of the follc'xing lemnra
9
Lemma
L,et
p
I
11r) be a rank (corank) function
of a graph G with edgc set E. Then, for all AgE,
thc
following hold:
lAl =nc(A)+p(A)
141 =
q(A) +.rr(A)
p(E) = nr(A) +P(A*)
s(E) = n"(AFlr(A*)
Proof
Let A, (respectively, 4) be a maximal subset of A which does not spntain cub€ts (circuits). Then
A\As (respectively,
any edge in
consequently
lA\Arl ( lA\D
A\)
forms a cutset (cfucuiD with edges
h A, (A")
only and
equals a maximal number of independent cutsets (circuis) of a gnph
G, made of edges in A only. But:
IAW = L{l-A(A)
IAW=lAl-p(A)
and therefue we obtain
jr(A)
(la)
n"(A)=lAl-p(A)
(lb)
n.(A) = lAl-
I€t pBgrB)
be a rank (respectively, corank) function of contraction (redrction) of G from E to BcE.
Then forall AC.E
[ll]
:
pB(A) * p(A
u
JrB(A) = Jr(A
(ErB)) - p(nn)
u tE\B)) - pt{Rts)
l-et A=B, then we obtain:
pA(A) = p(E) - p(A*)
(2a)
AA(A) = A(E) - $(A*) (?b)
On the other hand, for p and
A
the
follo*'ing relationships hold for all AgE, [l
lAl-s(A)=p(E)-p(A*)
lAl- p(A) = AG) - A(A*)
Cc'r;rparing (2a) wittr (3a) ad (2b) witlt (3b) we clrtain:
l]
:
(3a)
(3b)
pA(A) = lAl- A(A)
(4a)
AA(A)=lAl-p(A)
(4b)
Comparing (1a) wirh (4a) and (3a) and er,,mparing (lb) with (4b) and (3b) we conclude that the
following relations hnltl
l0
p(E)=n.(A)+p(Al)
(sr)
A(E)=n.(A)+A(A*)
(sb)
o
Proof (of Proposition 4)
Recall relations (la"b) and (5a"b) and qpecify A respectively as:
i) A=d
ii) A=d*
Obviously p(E) = rank G, A(f,) = corank G
0,
p(d) = ldl,
nc(d) = 0
nr(d) =
A(d) = ldl
and therefore we finally obtain:
i)
and
rank G =
p(d*),
corank =
A(d*),
Hl = corank d
ldl= rank d
ii) rank G = ns(d*)+ldl, rank d* = ld*l- n.(d*)
corank G = nc(d*)+ldl, corank d* = ld*l - nr(d*)
Corollary
I
tr
(to Proposition 4)
The sums Hl + n"(d*) and ldl + nr(d*) are invariants of a gnph.
Corollary 2 (o Proposition 4)
The numbers n (d*) and nr(d*) are invariants of the set of all doublc independent subsets of thc
same cardinality.
Note: to each class of double independent
d1 and
dze
subs€ts we may associate a
triple (d I n (dr), ns(d*)).
If
two double independent subsets of a graph and ldrl= ld2l + k then according ro
Proposition4, nr(Q*) = nc(dl* ) - k, andn (Q*) =ns(dt.) - k
3
@lusuating Corollaries I and 2)
Two copies of the same graph with wo different double indepcndent subsets of the same class,
Example
shcwn in figure 6. Note that
d, is a hyb'rid base while dt is not.
are
Nevertheless the staternents of
Corollaries I and 2 hold. Both d1 and d2an.e characterised by the triple (2 | 1,2).
Proposition
5
A subset b of edges of a gaph G is a hybrid base of G iff br is a minimal subset with the proirenies
rark b*= rank G and corank br,,=cortnk G in the sense that no other subset of b* has the samc
tirt'irrtY.
1l
Bold edger bebry
b doubL
lndcpcrdcnt rubsc!
ldtl=
(r)
2
Fzl= 2
o)
Figurc 6
The following algorithm for finding a hybrid base of a grryh is based on hopooitiur 5.
Algorithm
2
CIo find a hybnid base b)
Input: A graphG
begin
G'<_ G
G"e- G
b+-9
white corank G'=corank G and rank G"=rank G do
begin
Choose an edge ee b* that does not form either a selfcircuit of G' q
a selfcutset of
G".
Contract e in G'and remove e from G". Denote the graphs obtained by Ge'and G"" respectively.
G' e- G"'
G" G- G""
b e- bu{e}
end
end of Algorithm
g5
Hybrid
bases that belong
I
to a prescribed tree
in which a hybrid base is obtained by consecutive augnrentation of a
double independent subset" here we search for hybrid bases by making consecutive rr,'luctior",s of a
set of edges that contains a hybrid base as a proper subset. Evidently, any hybrid base is a sut-,set of
Unlike Algorithms
a tree and so
and 2
it is convinient to start with a circuidess subsct that contains
a hyb'rid base.
The dual
proposition can be fomrulated with cutsetless subsets. The main advantage in searching for hybrid
bases by making reductions of trees is in the following: it is possible to predict thc cardinality of a
hybrid base that belongs to a prescribed tree, before nue have actually found a hybrid base.
Moreover, it is possible to frnd a hybrid base u'ith prescribed cardinality by choosing a tree with
appropriate diameter.
t2
Proposition 6
I-et g be a circuitless subset of cdges of a graph having the following property:
(p) each edge in g* forms a circuit or a cubct with edges in g only.
Then for any edge eeg, the subset gt=g\{e} has property (p)
iff
the conjunction of t}re
follwing
two
statements holds:
i)
e belongs
ii)
any cutset that e forms
to to a cutsct made of edges in g only, and
with
edges in
gt
and any circuit made of edges
in g* only,
are disjrrint
Note. Since g is circuitless, the edge e does not fonn a circuit with edges in g\(e) only
therefore, according o tbc Painting thesem, e fomrs at least one cutsct with cdges in gt only.
and
Before we prove Proposition 6, we present the following lemma.
Lemma 2
Given an arbitrary subset g of a graph, suppose that eeg
that xeg* forms a
circuit
q
*ith
frms
edges in g only, then ee C*
a
cutset Se with edges in g* only and
iff xe S".
Proof
E g* and Cr\{x} g. g arc obviously disjoint and hence [e] c Srn C.C
SincelS"nC*l+ l,weconcludethatSrnQ = {*,e},thatiseeC* iff xeS".
The subsets S"\{e}
{x, e}.
tr
Proof (of Proposition O
+
Suppose that g1=g\[e]has property (p), then ee
gt* forms a circuit or
a cutset with edges in
only. Because g is circuitless, e does not form a circuit with edges in 91 only. This leads
o
91
the
followin g nro corrclusions:
a) e forms a cutset with edges h gr=${e} only,
b) e forms
a cutset
with edges in g* only
and
(acconding to the Painting theorem applied to the uiple (g1,
{e},g*)).Obviotrsly a) is equivalent to i). Supposenowthat condition ii)isnot tnre. Tharis,
suppose rhat there exists an edge xe
S.\{e}
c
g* that forms
a
circuit with, say C*, with edges in g*
only. Then according to the Painting theorem, x does not form a cutset with edges in g only and,
consequently, neither with edg''r in g\{e} only. According to pmp€rty (p) of g, x must form a circuit
with edges in g only. Denote this circuit by
fomrs with in g. Applying
lrmma
the edge e from g and add
C*.
Since g is circuitless,
C, is the unique circuit that x
2 to the pair (S",Cr), we conclude that
c
Cr. Now, if we remove
it to g*, ihis unique circriit is d.:stroyed Thus x both does not form a
cutset and does not form a circuit with edges in g\{e} only. This contradicts the assumption that
g\{e}
has
Foperty (p). Hence, cc'rrditlon ii) holds.
+= Suppose that the subset
Bt .' g\{e}do(.s not have pi,''i;irf} (p).
T?ren there exi'sts an edge
xegt*
that does not fonn either a circuit or a cu$et with edges in gt only. Fecause g r:ontains c and g is
,;-ircuitless, then e d*es not fonn a cii:r":itit with erlges in
g\{e} ,,81 only.
l3
Case 1. Supposc dso that e docs not form a cutset with edges in gt. Then condition
i) does not
hold and consequently ttre conjuction of i) and ii) is not tnrc.
Case 2. Supposc that e forms
I
cutsct with edges in 91 only. Then x*c. Because of
the
assumptions drat g has property (p) ana 81 docs not, thc edge x fqrns either a circuit or a cutset with
edges in g only such that either is destroyed by rcmoving e from g. We shall first show that if x
would form
a cutset
with edges in g only then this cutset cannot be destroyed by removing e from g.
Actually, if S, is a cutset that x forms with edges in g only, then either So contains e or not. If Sa
does not contain e then, after
rcmving
then because eforms acutset with Sc
9
witr
x still forms a cutset with edges in gr.
edges in 91 only,
(Sru S)\(Sou
ff
q)
So contains c
ir still
a cutsct
that contains edges x and edges in 91 only. Hence the assurytion that x forms a cuset with edges in
g only contradicts the assumption that 81 does not have property G).
At the begining of C:se 2,we claimed that the edge x forms only a circuit Crwith edges in g only.
Due to the assumption that g is circuitless, C, is thc unique circuit x fomrs with edges in g only.
Thereforc, C. can be destnoyed
iff e belongs to Cr. Thus ee C* and applying lrmma 2 we conclude
that x must belong to S"\{e}.
On the other band, according to the Painting theorem, if x does not form a cutset with edges in
only, then x forms a circuit C** with edges in
conclude that e does not belong to
C*f
gli.
91
Also, applying thc Painting theorem, we
because otherwise e cannot form a cutset
with edges in g,
only. As we pointed out in the note, e forms at least one cutset with edges in g* only. Thus we
finally have the following: both a cutset that e forms with edges in g* and a circuit that made of
edges in g* only (that is, C**) contain the edge e and therefore are not disjoint. This conmdicrs
condition ii) and consequently tlre conjuction of i) and
ii).
o
Proposition 6 enables us to develop a procedure for checking whether a tree contains a hybrid
as a prop€,r subset The procedure consists
a)
of consecutive applications of the following two
Given a circuitless subset g, find an edge ee g such that both conditions
base
steps.
i) and ii) of
proposition 6 are fulfilled-
b) eeg\{e}
The input set can be a tree of the gaph. The proc.edure terminales when either there are no more
cutsets of the graph in g or when there are such cutsets but none of the edges that belong to them
satisfies condition ii). There are threc possibilities:
1). The procedure carulot stafi because there are no edges for which conditions
i)
and
ii)
are
simultaneously satisfied For exa-mple t-his is the case with the graph of figurc 7a-
2)
The procedurc can start but in the final result we obtain a subset which is not double
ir dependent (fi gure 7b).
3)
The procedure terrninat;s with a doiible independent subset (figure 7c).
l.l
Start =
Finirh
Start
strn
J
Finirh
(hybrll bue)
finirh
(double independent subteQ
Figure 7
A tree may contain more than one hybrid base
hybrid bases (only 3 are indicated).
t
(a)
as in thc
bl
(b)
examplc of fig 8 whe,re there are 12 distinct
bz
b3
(c)
(d)
Figure 8
'Itre above procedure is formalizecl ln &e following algorithrn
l5
Algorithm
3
CIo frnd all hyb,rid bases that belong to a prcscribed
Input:A
graph G and a tree t
rec)
of G
begin
HeQ
Remove the edges in G that bclong to t and in the graph obtained find the set
form selfcutsets.
of edges that do nc
hnote this set by C.
Contract the edges in G that belong to
t*
and find the set of all trees
in the graph obtained" Denotc
this set of trees by T.
while T is nonempty do
begin
Choosc any trec
teT.
Find rhe union of all fundamental cutsets of G defined by edges in
t.
Denote this union by S(t).
if C n S(t) is nonempty then {t\t is a hybrid base} H<-Hu{t\t}
else {t\t is not
a hYbrid base} H<-H
T <- T\{t}
end
if H is not empty then output "H is a set of all hybrid
else output "there are no hyb'rid bases that belong
end of
bases that belong to
t'
o t"
Algorithn
Proposition
7
lrt b lx a hSrgi4l tmsc and let t be a tree of a graph . If bgt then lU = diamefer t = hl - ns(t).
Pr"oof
GivenagraphGandatre€t, letbbeahybridbasethatbelongtotasaproPersubsetThen
according o Proposition 3 and the fact that b belong to t, any edge in t\b fmrns a cutset wift erJgts
l,r b only. On the other hand b
itself is cutsetless and therefore b is a maximal cutsetless subset of t
Thuseach edgein t,bforms a cutset witli edgesinbonly andhencenr(t) = lfS l.
But lt\b l=h l-
fld
h l. According to Iffma 1 applied to A.=t, rank t*+nr(t) = lt l. Frorn the
o
lasttworelationsweobtain h l=rankt* (=diametert) = ltl- nr(t).
and therefore nr(t) = lt | -
As an immediate consequence of Proposition 7 we have the following corollary:
Corollary 3 (to Proposition 7)
All hybrid bases rhat belong to the same
tree of a gnph are of the same
caniinality.
l6
According
o Proposition
7, to find a hyhid base of prcscribcd cardinality c, we have !o apply
n c.The nirtrul cudirnliry in tle set of all lrybrtd bases of
not less tlen tlv minitnal dioneter in tlu set of all trees of tlat grryh.
Algorirhm 3 on a tnee of diareter equal
a graph
k obvionsty
$6 Hybrid bases and perfect pairs of trees
The notions of double independence and hybrid base are closely related to pairs of trees,
particulary to perfect pain of f,ees.
The first proposition in this section relates double independent subsets of a graph and pain of
tnees. The same result was implicitly enployed in tU in the p'roof of Theorem 11.
Proposition t
A subset of edges of a graph is a double independent subset iff it can be represented
as a set
difference of a pair of trees of the gaph.
Proof
=+t-etdbeadoubleindependentsubsetofagnphGandlet tbeatreethatcontaindasaproper
subser Because no any circuit of G made of edges in d only, the graph G6 obtained from G by
removing all edges that belong to d, has the same rank as G. Every circuitless subset of G6 is a
circuitless subset of G too. Hence, a tree t' of G6 that includes
M
as a proper subset is also a tree
of G with the same property. Consequently, (t,t') is a pair of trees for which fu' = d-
e [rt (rl,t2) be a pair of tr;es of a graph G. Then, t1\2 is a subset of both t1 and t2*. Thereforg
tf2
is a double independent subset of G.
Because each hybrid bases of a graph is double independent we conclude from Proposition 8 that
any hybrid base is a set difference of a pair of trees. The converse statement is not generally tnrc.
For example, figure 9 shows six copies of a graph and for each copy a differcnt subset of cdges is
indicated by the use of bold alges.
The subset of edges
and so is d2. JJowever, d1 is not a hybrid
d2=t3\r4, where
t1,\,t3and
t4 are
dt is a double independent subset
base whereas d2 is.
N*tice also that d1=t1\2 and that
all trees of the graph G.
The next proposition provides a link lxtween hybrid bases and perfect pain of trees.
Proposition
9
Irt b be a hybrid base of a S"ph. Then ths.re exists a perf.:,:t prir (t1,Q sucli that b = tt$
.
Proof
Because b is a double independent sribsr:i of the graph
exists a pair
r.rf trees
it follorvs from hoposition 8 that there
(tt,b,) such that b = tt\2. [}us tt\2 is
a t,;'lrr id barc e6 G
ar:d, ac,.ording part
l7
iii) of Proposition
a cutset
3, each edge of
with the elemens of
edges in
tt\
tr*nt2* cannot meks
circuits wittr edges itt
is complement (which includes tt*nt2*)
only. But
til2
together wi0r
cutsets with the edges
trV only and consequently
t$
tt*nb*
t1. Suppose that this is not true. Then, according pan
edge ee
b bt
ard hcnce 6c
rank tz* = ht\tzl. This means that tt is maximally
l, rank tltltt\|.
that for the case under consideration, equality must occur. That is,
tr*nt2*
belongs
only. Thereforc edges in ttrrrt2t malc
distant from the t2. On the other hand, according to Assertion
e'e
rnakes a circuit orland
We now provc
9 is also maximally
distant from
ii) of Assertion 2, there exists
an edge
such that a fundamental circuit with respect to t2 (defined by that edgc) contains an
ttntz. Consequently, t'2=(t2\e)u{e'} is again a Eee and such that
t1\t2
Ctt\'2. But subset
t1\'2 is, according to Proposition 8, also a double independent subset that contains (as a propfr
subset) the hybrid base
distant from
g
tfg which is a contradiction. Thus we have proved thar tt is maximally
and vice versa- Hence (tr,tz) is a perfect
d.2
pair.
t3
o
t4
Figure 9
Remark 2
The converse of Proposition 9 is not generally rrue. That is, if (t1,9) is a pe.rfect pair of rees then
their set difference is not necessarily a hybrid base. To see this consider figures l0 and 11. Figrne
10 shows four copies of the lame graph and within each a subsct of edges is indicated using bold
lines. Now (tr,b) is a perfect pair and (by inspection)
tr\
is a hyb'rid basr: while
t2\t
is
not
tt
rtlt2
Figure
Figure
1
12lrt
l0
I shows four copies of the same graph and again various subsets of edges arc indicatcd
using bold lines. Again (tr,tz) is a perfect pair while neither t1\t2 nor
t2\,
is a hybrid base. Tbc
marked edges (indicating tile2 rnd t2\1) fmm neitlrer circuits not cutsets.
rrb
r11r2
Figure
r2ltl
t2ltl
ll
The next proposition gives necessary and sufficient conditions for the set difference of a hybrid
pair of trees to be a hybrid base.
Proposition
Irt
10
(t1,t2) be a perfect pair of trees of a graph. Then
tl\
is a hybrid base iff each edge in t2\, thar
belongs to the fundarnental circuit with reqpect to t2, defined by an edge in tl*nt2*, forms a circuit
with edges in t1\2 only.
Proof
+
Suppr:se there exists an edge ee
defined by an edge in
to
tt*ntz*
j\t
that belongs to the fundarncntal r:ircuit with respect to t2,
and at the same time e forms the fundarnental chcuit Ce with resirect
tl in which at least one edge is in ttnt
. Thus, e does not
form a circuit with edges in t1\2 only
and at the same time forms a circuit Ce with edges in the cornplement of
the Pailting theci *m, applied to the triple ((
not fqrrm a cirtset with edges in
tt\ )\{e},{e}, t}2 ), we conclude that edge e does
tt\2 only. Since e, wlrich is an edge in the complement of t1\2,
does not form either a circuit or a cutset with edges in the cornplement of
of Proposition 3, tt\2 is not
t1\2 only. According to
a hyb,rid base
(proof by contradiction).
tft,
acconding part
iii)
l9
e
Supposc now that each edge io
defined by an edge in
tr*ng*,
tz\r
that belongs to the fundarnental circuit with respect to t2,
forms a circuit with edges in
that do rnt belong to the circuits that edges in
tr\
only. The rcrnaining edges in
h\r
tt*nt2* forms with edges in tz, do not belong to any
circuit made of edges in the complement of tt\2. Denote the set of such edges by R. Each edge y
that belong to R, according to the Painting theorem applied to the triple ((
necessarily forms a cutset with edges in
cutset
wi6
tr\2 only. Thus each
edge io
tr\ )\(V),{y},
t1\2 ),
tz\r forms either a circuit q
a
tltz only. On the aher hand, since dre pair (tt,b) is a perfect pair, accoding
3 all edges in tr*ng* formcircuia with edges in tlb only and all edges in trng form
edges in
to Assertion
cutsets with edges itr
tltz
only. According to part iii) of hoposition 3, tilb is a hybnid
basc.
o
As an immediate consequence of Propositions 9 and 10, we have the following propositioo which
gives both necessary and sufficient conditions for a subset of edges to be a hybrid base
Proposition 1l
A subset
i)
t$
of edges of a graph is a hybrid base itr
the pair of trees
(tt,td is a perfect pair of trees
ii) each edge in t2\1 that belongs to the fundamental circuit with rcspect to t2, defined by an edge
in
tr*nq*,
forms a circuit with edges in
tf2
only.
Remark 3
Condition i) and ii) of Proposition 11 are mutually independenr In other words neither (i)=+(ii),
nor
(ii)+f). To see that (i) does not imply
(ii), it is sufficient to consider thc exarnple of frgure 1l
Although the pair (tt,b) is a perfect pair, condition (ii) is not satisfied- C-onversely, in onder to sec
that (ii) does not
ir:ply (i), considor the example of figure 12. ln this example, by inspection,
condition G) ls satisfied but rank t1*=J>{:ltt\gl and hence (tt, te) is not a perfect pair.
Rernark {
Irt D denote the set of all tree diameters of a graph G, let P denote the set of all perfect pair
distances of G and let B denote the set of all hybrid base cardinalities associated with G. According
to part iv) of Assertion
is a subset of
D.
3, diameter ( tt ) = distance beween tt and t.2 = diameter (tz ) anA hence
P
On the other hand, according to Proposition 9, for the existence of a hybrid base
b in a graph, it is necesary that there exists a perfect pair of trees (tt, t2) such that b=tt\2.
Consequently, B is a subeet of P. Thus we have proved that BCP
rnin D
)
min P ) min B and max B Smax P S max D.
g D. It immediately follows
that
20
trk
Figure 12
g7 Hybrid bases of maximum cardinality
The concept of hybrid bases of maximum cardinality is clearly related to several important notions
in hybrid orientated graph theory. For example, it is related to maximally distant pain of trees, to
complementary pairs of rees and to opological degree of freedom- Hybrid bases of maximal
cardinality (as distina from maximum cardinality) have appeared only as an auxially notion
without a specific name (except in [4J where they are called dyads).
Proposition
L2
If (t1,t2) is a maximally
distant pair of trees then both
t1\2 and
b\t
ale hybrid bases.
Proof
Suppose that one of the subses
tl\
or
9\1 is not a hybrid
base, for example the subset
t$.
Then
there exists a hybrid base d that contains tilt2 as a proPer subset Acco'rding to hoposition 9 there
is a perfect pair of trecs (t'1,t'2) such that
httgl
t'f'2
= d. Becaus
tl S d = t'N'2,
we conclude that
( Hl = lt'r\'rl which contradicts the assumption that (tl,t2) is a maximally distant pair of
uees.
Corollary 4 (to ProPosition 12)
The card.inality of a largest hybrid
D
base
of a graph is not less then the maximal diameter of
the
graph. In other words, maxB 2 maxD.
Proof
Because a maxirnnlly disrant pair is always perfect pair, llie maximun'r tree diamcter in a graph is
r,.
equal to t6€ disi;..noe berw*een a maximally disii,rt lrsir of cs. But according to Froposition 12 the
set difference l-x:twe.en a maximally disrant pair of tr'res is a hybrid base and therefore the
cantinality of a largcst hy'crid base is riot less than the nraximal diamcier in the graph.
tr
2r
According to Remark
4,
max B S max D. Thus,we have shown that the following inequalities
holdsimultaneously: max B SmaxDand maxBlmaxD.ThusmaxB =rnax D.Thisimpotant
fact was fint pointed out by Sengoku t2l and recalled by Lin [3].
Remark
5
The converse of Proposition 12 is not generally tnre. That is,
both
tl\q
and
g\1 are hybrid bses,
if for a given pair of trees (t1,g),
then (t1,9) is not necessarily a maximally
disuntpairof recs
In order to see this consider figure 13. This frgure shows four copies of the same graph with
different subsets of edges indicated with bold lines. Now t1\t2 and
tz\t
e
both hybrid bases but
(tt,tz) is not a maximally distant pair of trees.
r2ltl
tll12
Figure 13
The next proposition is a consequence of Proposition E. It gives necessary and suffrcient
conditions for a subset of the edges of a graph o be a hybrid base of maximal cardinality. The
same statement appeared (withoutprm| as Theorem 15 in [4].
Proposition 13
A subset of edges b of a gaph is a hybrid
base
of maximal cardinality (a dyad) iff b is a set
difference of a r:axirrrelly distant pair of rees.
Proof
.+
kt
rrees
b be a hybrid base
(tr,
12) such thr;t
of maximal
canCinality. According to Proposition 8, there exiss a pair
b= t1\g. Suppose now that (tt, t2) is not a maximally distant pair of rees.
Then, for a maximally distant pair (t1', t2), which must exist, the set difference
(according to Proposition
assumption that b is a
.:
l2)
a hybrid base. But
hyhid base,:f
lbtl=ltt\'l>ltt\l=lbl
koposition 12, b is a hybrid
tt\'=bt
is also
which contradicts
the
maximal cardinality.
= Suppose that b is a set difference of a r', 'ximally disurnt pair of trees
(tt,
!).
Then acconding
o
base. Any other hybrid basc , according to Proposition 8, also is a set
difference of a pair of trees. But ht\rrl2
maximal cardinality.
of
ht\'l for all tt', b' in a graph and consequently
b is
of
22
The following proposition relatcs hybrid bascs to distinct and complenrntary trees of a
Proposition
faph.
L4
Given a gaph G, let b be a hybrid base and let ds, be a maximal double independent subset of b*.
If
lbl = rank G then b and
d, constitute a pair of distinct trees in G.
We prove the following lemma beforc proving Proposition 14.
Lemma 3
Given a hybrid base b of a graph , let d6. be a maximal double independent subset of b*. Then ds*
and b are of the
sam candinalitY.
Proof
Since b is a hybrid base, according to Proposition 9 there exists a perfect pair of trees (t1,t2) such
that b =
rilb. The subset g\1 obviously belongs to b*. Because b\r C tl*nt2,
independent Since (tr,tz) is a hybrid pair of trees, any edge in
h rz\r only and any edge in trnt2
Proof (of Proposition
tt*nr2* forms a circuit with edges
forms a cutset with *dges
maximally double independent in b*. Since hl\t2l =
lt"\tl,
b\r b also double
it tz\r only. Thereforc,
we conclude that lbl= Hu"
l.
t2\1 is
o
14)
Given a hybrid base b of a gnph , let d6. be a maximal double independent subset of b*. Then
according to l-emma 3,
4.
and b have the same cardinaliry. But the cardinality
of a double
base.
tr
independent subset is always less then
rankc and thereforc
4.
is a hybrid
As an immediate consequence of hoposition 14 we have the following corollary:
Corollary (to Propmsition 14)
Iet b be a hybrid base of a graph. Then the following
i) b* is a hybrid base
ii)
h*l=
lb
staternents are equivalenc
I
iii) (b,b*) is a complementary
g8 A1:plication
pair of u'ees
in hybrid network
analysis
'lhr: classical approach in hybrid analysis of electrical networks deals with, so called, topologically
complete sets of network variables. Such a set @nsists r:f a minimal nrtmber of voltages and
cu; re1ts whose valiies ar e sufficient to determine one of two variables (voltage or current) of every
,:.lge of the nr:twork, by n:e.ans of Kirchhoffs laws only, providing that only *ne variable (eiiher a
23
voltage or a current) of an edge can contribute to the opologically complete set In gfaph theoretic
terms, this notion can be formalized by introducing a pair of disjoint subsets (t,p) of edges such
is circuitless, p is cutsetless and each edge in the complemnt of a-4r either fomts a circuit
with edges in t only or forms a cutset with edges in p, only (see for example [2]). The set of
that
r
voltages of the edges in
r
and curents of the edges
in p then form a topologically complete
set
of
network variables.
The cardinalities of corylete sets of variables of a network may differ. Certainly, a complete sa
of the network with the minimd cardinality is of particular interesL As has been pointed out by
Kishi and Kajitani [1], the minimal number of network variables in hybrid analysis coincides with
the maximal distance betrveen trees and atso with the minimal hybrid rank of the gnph associated
with the nework. The minimal hybrid rank of a gaph was introduced by Tsuchiya et al. IQ and
called the topological dcgree of freedom of a graph. On the other hand Sengoku [2] first ob,senred
that the cardinality of a largest simultaneously circuidess and cutsetles subset (in our terrninology a
largest hybrid base of maximum cardinality) is equal to the smallest number of topologically
complete newort variables.
Given a hybrid base b of a graph G, let
t (respectively, [r) consist of all edges in b that belong to
the circuits (cutsets) of G that edges in b* form with edges in b only. Obviously, the union of
and p is equal to b. Ttre voltages associated with edges
in
t and currents
r
associated with edges in p
form a hybrid set of variables in the sense that these variables determine at least one of two
variables, voltage or current, for each edge in b*, by means of Kirchhoffs laws. This is clear
because b is both circuitless and cutsetless and becausc any edge in br forms a circuit with edges in
t
only or a cutset with edges in p only. If tnp is empty then exactly one variable, either a voltage
or a crurent is associated with each edge in b and therefore thc set of voltages in
form a topologically complete
topologically complete
set
of variables. If
t
and curents
p
tnlr is nonernpty then we have an "almost''
set of variables because then each edge
in
rgr
contributes to the hybrid sct
of variables with two variables: voltage and cunent The number of elements of such an "filmostn
topologically complete set of variables is equ*l to ld*tl + lp\d + ztr4rl. For any hybrid
number is always greatsr than or equal to lbl. Equality @curs iff
a
t
trase b this
and p are disjoint" that is,
iff
b is
hybrid base of maximum cardinality.
The following algorithm gives
base b.
a procedure
for finding a pair
(tg)
associatr.:d
with
a given
liybrid
21
Algorithm
(To find
a
4
pat (t,p) of subsets for a given
hybnid base b)
Input
A graph G and a hybrid base b.
begin
for all ee b* do
begin
find the circuit C" drat e forms with edges in b only
if C" exists do mark
edges in Ce\
{e}
as v-edges
end
for all ee b* do
begin
find ttre cutset S" that e forms with edges in b only
if S, exists, do mark
edges
in Se\ {e}
as i-edges
end
Denote by
t
the set of all v-edges and denote by p the set of all i-edges.
end of Algorithm
Proposition
15
Given a hybrid base b of a graph, let t (respectively, p) consist of all edges in b that belong to
circuits (cutsets) that edges in b* form with edges in b only. If the cardinality of b is less than
maximal dren each such pair of subsets (t,p) has a nonempty intersection-
Proof
t
and p are disjoint Then
t and currents of edges in p fonn s tr'i)ologically
complete set of variables.
Suppose rhat the intersection of
voltages of edges in
t
and p is empty, that is, suppose
This is clear because any subset of b is both circuitless and cutsetless and any edge in
the
complernent of the h;. i",rid base b forms a ' rircuit or a cutset with rdg':s in b only. Since the union of
t and p is equal to b and tlre intersection
is nonempty, we conclude ilrat the cardinality of b is equal
ro rhe cardinaliry of a topologically eoorpl*te set of variables which is greater than or equal to the
topoittgical degree of freedom. Hence the cardinaliry of b is greater than or equal io the topological
degree of fir:eedom ( hybri,l rank) of the graph . But, acuording to Rerrrark d, the cardinality of b is
always less than or equal f,;
{r" maxi'rr:l
,Jistance be
degree of freeedunl which is a contradicl;tm.
tween pairs of mees, that is, to trc topological
25
A main goal of this section is to show that the number of hyb'rid nerwork variables can bc madc
smaller than the topological degree of frecdom by using hybrid bases of non-maximal cadinalities.
For this purpose, we will relax the definition of the topologically complete set of variables by
omitting the assumption that an edge can contribute in the hybnid set of variables wittt cxrtly onc
variable (either a voltage or a current). Thus we shall consider an "almost" topologically cunpletc
set of variables. Although such a situation with nvo independent variables associated with a
nerwork edge is physically admissible (for example in case of norator's edges in a nctrrork),
nomrally the voltage and the current associated with an cdge of the netrrork are mutually tied up
with the so-called constitutive relations. Briefly qpeaking the constitutive relation of a l-port is
described by a collection of pairs of signals of voltage and currents that are allowed by thc l-porr
In the case when the l-port is v-controlled or i-controlle4 one of these nvo variables (voltage or
curent) can be found from the other and consequently, the total number of nerwork variables
needed to be associated with each edge is one. Thus when the total number of variables is equal to
the card.inality the union ru4r, that is, to the cardinality
of
the hyb'rid base b.
But lblis always
less
than or equal to the opological degre of freedom and ttrerefore the total nuber of variaHes is less
than the topoplogical degree of freedom-
As an illustration, we consider an electrical network with the six resistors whosc constitutive
relations are
:
v1= v1( i1)
(cR1)
v2= v2(i2)
(cR2)
ir= i3( v3)
(cR3)
ia= ia( va)
(cR4)
ir= i5( v5)
(cRt
v6= v6(
(cRO
5)
Irt G be the graph (asscyciated with the network underconsideruion)
ro see that (rank G)=(cormk G)=(hyb'rank G)=3.
Irt
shown in figure 14. It is easy
b=(3,6) be a hybrid base of G of
nonmaxinral cardinality (bold edges). Ttre edge orientations are related to the edge currens and thc
edge voltages. The singly marked edges 11,2,61 indicate i-controlled resiston and the doubly
marked edges {3,4,5} indicate v-c*rrutrlled resisors.
Figure 14
In this case, using Algorithm 4, we obtain T= F = b. Hencetnrlt * b and hence I v3, i3, ve,kl
constitutes a hybrid set of variables.
25
Doubly marked edges form circuis with edges in t 1=6; only and singly mnted edges form
cursers with edges in p (=b) only and consequently ttre associated Kirchhoffs laws are:
Kirchhoffs current laws (KCL)
ir-ir*k=0
(KCLl)
it+ir-i6=0
(KCL2)
Kirchhoffs voluge laws (KW)
-v3+v4fv6=0
Ga/Ll)
-v3+v5*v6=0
(rryL2)
Substituting KCLI, KCL},
KVLI
and KVL2 into CRl, '.., CR5 and CR6, we obtain each
voltage and current of the net'work uniquely expressed in terms of the variables v3, i3, v6 and i6:
v1= v1(
(CR1+KCLI)
i:- id
(CR2+KCL2)
v2=v2(-i3+ i6 )
(cR3)
ir= i3(v3)
ia= ia( v3 - v6 )
(CR4+KWl)
i5= i5( v3 - v6 )
(CR5+KW2)
v6= v6(
(cR6)
16)
If we substitute (CR3) and (CRO injo
the other relations, we finally obain:
v1= v1( i3(v3) - 5 )
(CRl+KCLI+CR3)
v2=v2(-i3(v/+g)
(CR2+KCL2+CR3)
(cR3)
ir= i3(v3)
ia= ia( v3 - v6( ip )
(CR4+KWl+CR6)
i5= i5( v3 - v6( g) )
(CR5+KVL2+CR6)
(cR6)
v6= v6( i6)
Writing I VL for the circuit that edge 3 forms with cdges I and 2, and KCL for the cutset that
-:dge 6 forms with edges 4 and 5, and expressing all variables in these equations in terms of
variables v3 an6 i6, according to the last equations, we finally obtain [bl=2 equations in terms of 2
'rariables {v3, i6}:
k) f v3 = 0
ia( v3 - v6( ip ; + i5( v3 - v6( id ) - k= 0
vr( i3(v3) - i6 )
+ v2( -i3(v3) +
Thus the number of equations and the number of independent variables is less then the topological
degree of freedom, which we intended to show'
27
$f0 Concluding remarks
In this paper we introduced the new conc€pt called hyb'rid base of a graph. Several propositions
were stated in order to closely characterise its properties and to relate this concept to some
important notions in hyb'rid oriented graph theory. Fcr example it is related to maximally distant
pairs of trees, to perfect pairs of trees, !o complementary pairs of trees, to hybrid rank and to
topological degree of freedom. Ir is also related to the problem of finding the minimum number of
independent variables in electrical networts. Also, several exarnples are included to help the reader
gain intuitive insight. A complete list of hybrid bases of a number of graphs are shown in the
appendix.
References
tll
I2l
Kishi G and Kajitani Y, " Maximally distant trees and principal partition of a linear graph"
IEEE Transactions on circuit Theory, vol.cT-16,323-330, 1969.
Sengoku M, "Hybrid trees and hybrid nee graphs", IEEE Transactions on Circuits
t5l
and Systems, vol.CAs-zz' 78G790, 197 5'
Un p M, "Complementary trees in circuit thesy",IEEE Transactions on Circuits and
Systems, vol.CAS-27 , 921-928, 1980.
Shunguan G and Chen W-K, "Tbe hybrid method of nerwork analysis and topological
degree of freedom", Proceedings of IEEE International Symposium on Circuits
and Systems, Rome, 158-161, 1982Amari S, "Topological foundation of Kron's tearing of electrical nerworks", RAAG
t6l
Memoirs, vol.3, 322-349, 1962.
Tsuchiya" T; Ohrsrki T; Ishi"ah Y; andWatanabe
t7l
Ohtsuki
t3l
t4l
It Kajitani Y; and Kishi G,'"Topological
degrees of freedom of electrical networks", Proceedings of the 5th Annual Allerton
conference on circuit and system Theo4r, 644-653, 1967.
f, l5fiirki Y, and Watanabe H, "Topological
of freedom and mixed
analysis of electrical networks", IEEE Transactions on Circuit Theory, vol.CT-I7,
dcgrees
491-499,1970.
t8l
t91
Novak L and Gibbons A, "Perfect pairs of trees in gtaphs", International Journal of
Circuit Thc.ory and Applications, to appear.
Deo N, "A cenrral trec", IEEE Trans. on Circuit Theory, vol.CT-l3, 439-M0,1960.
t10l Welsh D J A, Manoid Theory, Academic Presq l-ondon, 1976'
tl
U
Swamy M N S and Thulasiraman K,Grairhs, Networks, and Algorithms, John Wiley &
Sons, New
Yorlq
1981.
Chua L O, "The colored branch theorem and its applications in ciicuit
Transactions on Circuits and Systenrs, vol.CAS-27, 816-825, 1980.
I ad
t12l
Vandewalle
t13l
theory", IEEE
Gibbons Alan, Chapter
1985.
I of "Algorithmic
Graph Tl,€or)", Cambridge Univenity Press,
2E
Appendix
All Hybrid
Bases
for the Graph of Figure
3
a) with cardinality 5 (8 items)
I
3
3
23 6ll
3 6910
3 69ll
36 1114
6 8 1114
69 1014
36
36
l4
12
l4
7
9
11
l0
14
13
b) with cardinatity 6 (80 items)
t34 7
r349
1357
135 9
r 36 9
136 9
13 610
| 46 8
| 46 9
l56E
156 9
234
2349
23 5 8
235 9
2369
23610
23610
246 9
2469
256 9
256 9
3 67 9
36710
368 9
3 6I 9
3 6 810
E
11
10
14
13
I
I
I
l4
l0 l3
l1
t2
t4
t2 t4
ll t4
10 t4
11
I
I
I
I
ll l4
10
l4
l4
l4
11
t4
1l
r3
l3
12
14
ll
1
I
2
2
2
2
2
2
l0 l4
ll
10
ll
10
2
2
2
2
3
3
3
3
3
3
t2
t4
t2
l4
t2 l4
1l
1l
1l
13
t2
E
1
13
10
34 E
34 9
3 5
3s 9
3 6 9
3610
3 610
46 9
4 6 9
1
r3
12
14
14
s6
s6
9
9
9
9
9
9
9
34
34
3s
35
3 6
3 610
4 6 7
46 9
56 7
56 9
67 9
67 9
6710
6 8 9
6810
6t 10
1l 14
10 14
lt 14
l0 14
1l 13
lt 12
t3
t4
l0
ll
t2
t4
10 t2
11 14
10
11
12
14
12
14
t0
11
t2 t4
11
ll
t0
11
l0
13
14
13
14
13
nt2
t3 14
t2 t4
11 13
11 12
13 14
9 1012
9 u14
9 r012
r 35 9 nt4
136 9 1214
r 36 10 ll 13
134
134
135
| 46
| 46
r46
156
234
234
235
235
236
236
236
246
246
256
256
367
367
367
3 6I
36t
7
9
7
9
7
9
7
I
9
9
10
E
9
1114
1014
ll14
to13
ll 14
1013
1l14
1013
1112
13t4
12 14
1114
l014
8 n14
9
9
10
10
9
10
1014
1lt3
n12
13 14
1214
ll
13
c) with cardinalitY 7 (128 items)
l3 47 9
13 47 9
l3 4710
13 4 8 9
134810
134810
9
1357
135 710
ll
13
12
ll
ll
13
12
12
14
14
I
I
I
13
I
12
14
14
I
n13
I
I
I
34 7 9
34 710
34710
34 8 9
34 810
35 7 9
35 7 9
3 5 710
1l 13
ll 12
13 14
t2 t4
lt 13
lt t2
13 14
12 14
13
13
13
13
13
13
13
13
4
4
4
4
4
5
5
5
7 9
7 10
t
8
8
7
7
7
1214
1113
9 llr2
9 t314
l0 1214
9 ll13
l0 l112
10 1314
29
135
135
135
t46
| 46
| 46
146
146
156
156
156
156
89 11 12
89 13 L4
810 12 14
79 ll 13
710
710
89
810
79
79
710
E9
t10
r 56
s6 t10
r234
79
234 710
234 t9
234 89
234 810
235 79
235 710
235 710
235 E9
235 tl0
246 79
246 79
246 710
246 89
246 tl0
246 tl0
256 79
256 710
256 t9
256 89
256 El0
1l t2
13 14
t2 14
ll 13
1r 12
13
t2
lt
l1
13
12
lr
ll
13
12
11
13
t2
t2
I
I
I
1
t4
1
13
12
14
14
13
12
14
t4
14
13
12
14
t4
il13
ll 12
13
I
14
13 t4
ll
1
I
I
I
ll 13
lt 12
13 14
t2 t4
ll 13
ll 12
t2
I
I
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
t4
2
2
14
2
35
35
35
46
46
46
46
46
56
56
56
56
56
34
34
34
34
34
34
35
35
35
35
35
46
46
46
46
46
56
56
56
56
56
56
E 9 1l
8 10 l1
810 13
7912
7 l0 l1
8 9 ll
8913
t1012
7 9 ll
7 10 11
7 l0 13
89r2
t l0 11
7911
7913
71012
89ll
81011
810 13
79t2
7 l0 ll
t 9 ll
E913
81012
79ll
7 l0 ll
7 10 13
l3
8 l0 11
7911
7913
71012
8 9 1l
8 10 ll
810 13
l3
t2
l4
t9r2
t2
l4
l4
l3
I
I
I
12
l4
l4
l3
t2
l4
l4
l3
t2
l4
t4
l3
12
t4
l4
l3
l2
l4
l4
l3
t2
t4
l4
14
l3
12
l4
I
I
I
I
I
I
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3589
3 5 810
4 6 7 9
4 6 7 9
46710
46 8 9
4 6 810
4 6 810
5679
5 6 710
56t9
56E9
5 6 E10
3479
3 4 710
3 4 710
3489
3 4 810
3579
3579
3 5 710
3589
3 5 8r0
3 5 tlo
4679
4 6 710
4 6 t 9
4 6 89
4 6 tl0
5679
5 6 710
5 6 710
5 6 89
5 6 810
t2
ll
ll
13
t2
14
13
t2
14
t4
l1 13
l1 t2
13 t4
t2 t4
1t
13
13
12
14
ll
t4
t3
ll
t2
1l
13
12
L2
14
L4
u13
l1 12
13 t4
t2 t4
ll 13
ilt2
t3 t4
t2 t4
1l 13
ll
t2
13 t4
t2 t4
ll
ll
13
t2
13 14
12 t4
tl l3