GRAPHS, GEOMETRIC REPRESENTATIONS
A ND B INA RY RELATIONS
L. O. KArrsoFF and L. HART
1. Introduction, It is we ll known that binary relations may be
represented b y graphs [ l
] . N o w , o f f in it e graphs, t h e re a re g e o me tric re p re representations
j u s to f b in a ry relations. I n th is p a p e r w e e xp lo re th e
sentations
a
relationships
s
between b in a ry relations, t h e ir graphs and th e ir
geometric
t
h
erepresentations.
r
e
a
r
e
g 2. Graph-Theoretical
e
o
m
e Concepts. Let r = (11„ v
set
2
o
f
elements
called
" ve rtice s" . The subscripts I , 2, n are
t
r
i
c
simply
, v labels
„ ) to distinguish
b
e
the vertices.
a
Let e = { ( v
pairs
i o f elements o f -r, ca lle d "edges". We denote ( v
i ,v
, vSince
i
v
iedges.
i) y
) DEFINITION
, bn y
a
1. I f v
e
d
end-points
1
o f the edge. v
iterminal
iva
, sv n
t end-point.
h e
id
i n
i i t i a l ,2. I f (v
DEFINITION
v n id se i)a
then
p ,rja ov in
(i) nv t ,
1
/
a ie
e d(ii)
k n gv i e) is, incident to
.
n
d )t
i ih ss e 3. I f T c e, then (ei,i e ( ( v i , Ili) a T and
DEFINITION
}
o
v an
v
ni
(v tb
i eT.
i
in
e
i e dn g
i an
a
n
s
c ni
d4. I f
, e
,
ef}, 1 is a graph.
td c
t DEFINITION
h
v
d
v e
5. G is a subgraph (also called a graph) o f g
e DEFINITION
vh s
i kse a e
if and
o
n
ly
if
G
E ) , where O * V g_ - r and E
i
) =rs i n
a
r 6 . A graph G is a model of the binary relation
t
T vDEFINITION
le
ythe set S if and o n ly if
e
R
on
t
) id
tt
o
o
. io
h
r
(
I sf
e
v
f vto
k
i
e ir
=
,
i n
d
v
v
j ce
i
T itr
,
)
, ,e
,
w4
d
a
e '
468
L
.
O. K A TTS O FF A N D L . H A R T
(i) th e re is a 1 - I ma p f o f the elements o f S onto the ve rtices o f G such that (s
i (ii) th e re is a I I ma p between the predicates defined fo r R
and
, s the predicates defined f o r G, and
i (iii) e ve ry true statement about R translates into a true statement
about
G.
)
R
i f f
( f
( 3.
s Geo metric Representation o f Graphs. I n t h is section we
construct
g e o me tric representations o f graphs. A g e o me tric
i
representation
) ,
f ( sis a plane figure. Th e plane ma y be Euclidean
ior non-Euclidean.
) EXAMPLE
)
7. Figures I a n d 2, b e lo w. a re geometric re p re G
sentations of the graph G = f { v
,
vE
i
representation of G, we first select any three
,1struct
v • ,a, geometric
v
)3
points in the plane. To each p o in t we correlate one and o n ly
1
)one
, vertex.
( v In Figure I , we use the map g such that g (v
ii (v
1
g
.2
that
) —g ' (v p
,
v
v)i
iT
ii
)h=
Pi
, in
= Figure 1 a n d p '
)lp
( v
a,p(v'i ,n
2
t)g
1
F
23 (vi g u r e
,2
v sh o wn t o b e in G b y th e d ire cte d segments f ro m
ia, g . a re
,iedges
7
iW
isg
)sn) ' (v
e
)s(,show
vof the segments.
da v , the
ta
2
h direction
o (
n
V
2
w
3
gr
)o
e
(e
tg
h
a
,t)—
fo
t=
ve
(d
vg
(p
v3d ' v
P'
'le
2
(i)g v
3
P
,2i)=e
v
'
2
n
v)a
ia
-s
Pg
Figure
I
F
i
u
r
e
2
G
2a
)n
pi
n
'
t
,,n
d
a
3n
d
If
there
we
re
a
bi-oriented
edge
in G, th is wo u ld be shown
w
vd
g
n
.G
g
by
a bi-directed lin e segment in the geometric representation,
/3g
d
Ib
(e
i.e..
a lin e segment with o u t an a rro w.
(1'vd
ny v
ar(3
3
F
d
)nv)a
ir
d3
=
ga
tw
a
E
p
uw
)o
'=ctg
ri
eo
3
(ien
.lg
vr2g
lic,t
T
GRAPHS, GEOMETRIC REPRESENTATIONS
4
6
9
Since there a re six d istin ct maps o f th e vertices o f G onto
the points in Figure I , the selection of a set o f three points in
the plane i s n o t su fficie n t t o determine a unique geometric
representation o f G. Furthermore, since we can select any one
of an in fin ite n u mb e r o f trip le s o f points i n th e plane, th e re
are in f in it e ly ma n y possible g e o me tric representations o f G.
Since the remarks are general, we conclude there is no unique
geometric representation o f a graph G.
DEFINITION 8. T w o graphs, G a n d H, a re iso mo rp h ic i f
and o n ly if there is a 1 - I ma p f of the vertices o f G onto the
vertices of H such that (v
i DEFINITION 9. T w o g e o me tric representations, L a n d M ,
,arev isomorphic if and o n ly if there is a 1 - I ma p of the ve rte x
ipoints o f L onto the vertex points o f M such that
)
E G
i (a)f p
a i nis a ddirected line segment in M,
o p n
l
y
i i(b) p f
segment in M, and
( ii f s is( a bi-directed
v
p
,
i a(c) th e re is a circle at p
) dii , si at
n f (p fL
i f
a rv i n
( ia
d
THEOREM 10. I f R is the set of all geometric representations
i) n
i eb
l then,
y if r, and r
of o
a graph
G,
) ci i n f E
i a r e o f aR), nr y
elements
H ttd i Mh .
r
it Proof.
i sw
o e
re . Since r
e
ir sie oapam
there
nr odI es-r Ismae p sn f, t a nad f
iih oi cre
de
tvertex
p
n t s h in er, and r
i rcao tf points
tj(v li e i ro t .i c e
.
e
irte x points
r e sco f r
of
thei rve
r.,o iacS
n ef c ee
f
ld
iiG n
oo s
directed
in r
gso neit segment
t.o ie
h
e
iI(f
f
a
n
d
m
o
e en t r t vt si en c rl h
t y i c ee s
o
1—
g
1
ioio e
rm nfe fpt r (o f ,
rected
g
lin
e
segment
i
n
r
(
G
.
—
fl ee s e
iiand
m
p
(N n bi-directed
l (o p segments i n r
—
nS it m ai l ta r
.map
e
i
iip
w is oh
or ron w s p t h a t
t
o n s fi d e r a
)
if)lc,i n
oi ,
tp
t
i
o
n
s
f
i—
o
n
t
i) fn
s
fio
i
o
r
1
fa,i G
7 r
'
(
p
L i
c
n
r
c
f
i-jIfi ,1
i
s
i
s e
7
,t)-l7
s L ) eh
f
i
1
id
a
.
e
s
i
r
e
I1 a
f
(
s.d
( ,
m
a
n
a
p
tipp
d
i
h
ni n
470
L
.
a K A TTS O FF A N D L . H A R T
shown t h a t iso mo rp h ism o f graphs a n d hence o f g e o me tric
representations a re an equivalence relation.
4. B in a ry Relations.
DEFINITION 11. I f A and B a re sets, th e n A X B = ((x , y):
x E A and y 1 3 ) . A X B is ca lle d th e Cartesian p ro d u ct o f A
and B.
DEFINITION 12. R is a b in a ry re la tio n o ve r a set S and o n ly
if there e xist sets A and B such th a t
R g { ( x , y ) : x e A and y B and A S and B g s }.
DEFINITION 13. I f R is a b in a ry relation, then
.9(R) =
2 (R) is called the domain of R.
DEFINITION 14. I f R is a b in a ry relation, then
M(R) = { y : ( x , y ) R } .
M(R) i s ca lle d the range o f R.
DEFINITION 15. I f R is a b in a ry relation, then
..F(R) = f x : ( x , y R V(y,x)G11.).
,9
range
and domain o f R.
( THEOREM 16. I f R i s a b in a ry re la tio n , t h e n t h e g ra p h
R
G = {5
) 4
17. A graph G is re f le xive i f and o n ly if, f o r
i( RDEFINITION
) , Rve rte x v, o f G, ( v
every
s}
ic DEFINITION 18. A graph G is symme tric if and o n ly if, f o r
i, vs
a
every
ve rte x v, and v
a
iis
ilm an edge in G.
),l ( vi s
a
n
o
DEFINITION
19. A g ra p h G i s t ra n sit ive i f a n d o n l y i f
e
d
g
e
ie
d
whenever
i,d v n (e,,EG and e
e
l
iGkTheorems
. 20 and 21 f o llo w e a sily f ro m the definitions.
itE
o G),
t h20.e I f nthe graph G is a model o f the b in a ry re THEOREM
)h
i
s
(flation
e ,R,
then
e
a
n
kR (a) R is re f le xive i f f G is re fle xive .
fG ) d.
g
e
.e
is symme t ric i f f G is symme tric, a n d
i (b) R n
e (c) R is tra n sitive if f G is tra n sitive .
G
il
m
p
l
id
e
s
(o
v
if
,R
v
i.
)T
h
GRAPHS, GEOMETRIC REPRESENTATIONS
4
7
1
THEOREM 21. A g ra p h G i s symme t ric i f and o n ly i f a ll
edges in G are bi-oriented.
THEOREM 22. I f G is a fin ite graph, then G has a geometric
representation in the plane.
Proof. L e t ( v
denote
i
the map o f the vertices in to the Euclidean plane such
, vf o r e ve ry i, h (v ) = (i, V (1-i)(i-n) I). The function h maps
that,
the2 vertices onto points on a semicircle in the upper half-plane.
, sevmicircle has radius (n -1 )/ 2 , and its center a t ( ( n
The
0).
- n
)
denote h (v ) b y p
1 We
b
e representation of G as follows:
geometric
i
t
h
1 ).(a)
/ 2 ,W
i f ee
cei o m p l e t
s, i s
e
tea n h
e
tce o n s t r
od c t i o n
u
fg
o
e
fve
i
f
e
rt(b)
t
h
i
from
ie
p
in c
i et(c)
,Goi i sf se
p oi, nconnecting p
ment
,G
i fDEFINITION
a n d
w
23. I f R is a b in a ry re la tio n and P is a geo,metric
a
RG
e . n representation o f ( g
a
n
presentation
d
of R.
. .d
n
d
e
L
r ) , R ) ,24. I f P i s a g e o me tric representation o f t h e
( RTHEOREM
d
eie
ahi erelation
tbinary
n
11, then a ll lin e segments are bi-directed in P if
iw i
a
t
P
and o n ly if R is symmetric.
i hria
s Let G = ( . 9
Proof.
sc
e
a
.
G is symme tric if f (e
n
b
g
o1 1 m) . e
tw i l l co n ta in o n ly bi-directed segments.
l i( Re) , representation
metric
o
r
rOn
i
c
the) other4i hand,
, RG
4s wh e re f is the appropriate map o f . 9
t
ein the
7
y m
m plane,
e (r cs epoints
the
a bi-directed segment connecting p
,
h
(i„ Rtla
p,
sh
) nor tws
i
o
t
c
h
a
t
b
o
t
h
(
f
d
iw
-G1e
edges
i
in G,f i.e., that G is symmetric.
e
n
(apa
a n
nd
d
G
1
e td
,rp
)i , o
f
n
a
w
i
-, -l 1G
)y .
w
e
(T p,i , ) h)
u
t
d
a
d
s f n,
rh
(t G
f
h
e
a
-e i1
d
w
(g sp
e
472
L
.
O. K A TTS O FF A N D L . H A R T
THEOREM 25. I f P i s a g e o me tric representation o f t h e
binary re la tio n R, then R is re fle xive i f and o n ly if there is a
circle at each vertex point of the representation.
THEOREM 26. I f P i s a g e o me tric representation o f t h e
binary relation R, then R is tra n sitive if and o n ly if
---->
(a) i f p
i
(b)
p if p
ii
(c)
if p
p
-a
ii n
(d)
i
f
p
d
p
n
THEOREM
ia
27. I f R and R' are two b in a ry relations such that
p
id
.F(R)
p
ia n= g (R'), P and P' a re the geometric representations o f R
p
and
ip R', and P is isomorphic to 1
d
ik n L e t f denote the isomorphism f ro m g (R ) t o P, g the
3 1Proof.
a
p
p
, d
isomorphism
t h e n f ro mR . F(R
a
ik
i rp
1
to
P',s then g
p
a
-)i - 1EXAMPLE
iest oh
o f mP o'28., r I tpis hmistake,
i c a lb e it a natural one, to believe
kr
that,
under
the
conditions
stipulated in Theorem 27, R and R'
t p
ia
s
n
d
o
ia
can
be
shown
to
be
identical.
Th a t such a vie w is mistaken is
R ke
h
a
n
'
.
n
ri
shown
b
y
the
fo
llo
win
g
example.
it P
a
s h
o m e
e
n
i Let
o
r, sr p
Ro =h ((a,
m b),
o (a,
r c), (b, c)) a n d R' = ( c , a), c , b), (b , a)).
iP fie ld of R is the fie ld o f R', and R is the converse of R'.
The
ip e
s
h
m
i
s
m
tn
f i,h
r
P
t
ro e
n
m
,
P h
o
P
n
t
m ,e
p
h
R tn
ie
t p
h
p
n
Figure 3
o ie
kp
p
Rn
ii
ks we use the map f such that f(a) p , f(b) = q and f(c) = r,
' Ifp
p
. ii, Figure 3 is a geometric representation of R. I f we use the
then
ks
p
map
g such that g(a) = r, g(b) q and g(c) p , then Figure 3
in
i
k
is P
a geometric representation o f R'. Cle a rly, Fig u re 3 is iso sn
i,
morphic
t o itself. Thus, th e conditions o f Theorem 27 obtain,
butiP
sR R ' .
n
,i
P
n
,
P
a
.
n
d
GRAPHS, G EO METRI C REPRESENTATIONS
4
7
3
5. Distance and the Properties of a Bin a ry Relation.
In th is section, w e ta ke th e concept o f distance f ro m one
vertex v, to another vertex v
i i properties
the
n
a
of a binary relation fro m its geometric representation.
g r a p h ,
a
n
d
DEFINITION
29. Suppose v
u
s
e
G, and
i aQ nis da setyof ,edges o f G. Q is a chain o f length n fro m
iv, to av, iftr and
e o n ly if there is a 1 1 ma p q of (1, 2, o n t o
tQ such
or t i c e s
v ethat
i (a)do[q (m)
e = f nv
t
i
f (b)
yth e re is a Y, such that 11(1) = (y
t a
i) gthae renr is da va
(c)
p
t,q is
,u )m
c, h athatn if there
d
It
hys( evident
is a chain of length n fro m v, to v,
in t+G, hthen
t
is a min ima l chain f ro m v, to v
1 a) there
i q
chain
, sh io rt. as
e any
. , other froam v, to y . We use such
=i n thatGis as
min
ima
l
chains
to
define
the
distance between a vertex y
( vn
)
i ve
n, xdvv, in G. We use d ( v
a
=
„a rte
to
i (vv y
iv , DEFINITION
t) )t o
30. Suppose v, and v, are vertices o f the graph
.G.
d ,—
e n ov t e
Then:
t i›(a) hi f (v e
d )((b)
i i i f.s( v „t v a n c
e y(c)
i, vi f v, and v, a re not coincident to an edge in G and there
f it) i isrs a min o
m C of length n fro m Y, to v
ima l chain
v =
n,
and
) n
a
i
i v(d)
i si,df vt h e n d ( v
e
a is
a
ig
ie no chain fro m v
)i n ivproofs
The
n
a
t, o o fv the
, fo llo win g theorems f o llo w easily.
e i n 31. v
d
n
THEOREM
d (i.e.,
) ne d ,
iyG
a,G
g
a
h
e
n d (y
v i ,t
rae
d
(
v
n, r e
i
THEOREM
32. A graph
G is symme t ric i f and o n ly if , f o r
ii a n in c i d
e
d
every p a ir of verticesy ,v)
n
v d nv t, )
+
i e
a n d
v33. , A graph
THEOREM
G is reflexive if and o n ly if, fo r each
o
c
iGe o
d
t
i
n v
vertex
t,d i .
G
d ( v ( y
i o
i n , v
THEOREM
c
34.
A gi ra p h G i s tra n sitive i f a n d o n ly if , f o r
(td
,
h
iG ,
o
,e( a vi v
v
vd ( v
n
,i r i ) i
b
ii
cv=e i r
)
o
i
e
)v
in b t
d
1
2
e
=
d
i e
,d) o
.
v
)( n
=t t
e
i1
t1
v. d h
,
o
i ge
a
474
L
.
O. K A TTS O FF A N D L . H A R T
every trip le of distinct vertices v
d
i (v, v
k, EXAMPLE
y ,
a n35.dConsider
y
th e g ra p h whose g e o me tric re p re )k
sentation
is as shown:
=
,
i
f
d ( v
Q
Ii ,
Ps
tv
P4
h
i
e
)
=
n
1
d
a
n
d
( v
i
P:1
v2
k
) This represents the graph G = ( ( v
=( ( vi
Ii( V 3, , VN2 ) r, ()V,4 , Vv4 ) , ( V 5 , V 5 ) , ( V 4 , V 5 ) , ( V 5 , V 4 ) , ( V 6 , V 6 ) ) ) . G i s r e .flexive,
, 3
symme tric and transitive. That is, if G is a model of the
relation
v ,
v R, then R is an equivalence relation.
i 4
THEOREM 36. A n y complete plane p o lyg o n o f n b i-d ire c) ,,
tional
sides
v
w i t h a circle a t each ve rt e x a n d b i-d ire ctio n a l
(diagonals
5
is a geometric representation of a strongly connected
y ,
v
equivalence
relation R o n a set S o f n elements.
2 6Proof. L e t S ( 1 , 2, T h e desired re la tio n is S X S.
, ) ,
v 6. E u le r Graphs a n d Eu le r Relations.
2 DEFINITION 37. A graph G is connected if and o n ly if, f o r
)every pair of distinct vertices v
,ito av, nin dG, ovr a , chain
fro m IT, to y
,
(ti h e r e
DEFINITION 38. Th e g ra p h G = (V , E) i s a n E u le r g ra p h
vi.
if and so n ly if there is a subgraph G' of G such that G' ( V , E'),
3
asuch that
,c
a
i
n
(a) hE
vf
r
o
m E' is a chain from Y, to y
i(b) f o r each Y, in V,
3
v (c)
,
ii s E ' is asymmetric.
)E' is called an Euler path in G.
c, a n d
,
o
(
n
v
n
i
e
,
c t
y
e
2
d ,
)
,
GRAPHS, G EO METRI C REPRESENTATIONS
4
7
5
DEFINITION 39. G — (V , E) is a directed Euler graph if and
only if E is an Euler path in G.
DEFINITION 40. Suppose t h e g ra p h G i s a mo d e l o f t h e
binary relation R. R is an Euler relation if and o n ly if G is an
Euler graph.
EXAMPLE 41. L e t S f s
(s i
3 , s
, 2
s , s
1 s
) )
a
n
d
1 R
. =
Pi
P3
A(
(
s
i
,
g s
2
)
,
This is e vid e n tly an Euler graph. I f we we re to add an elee (
s
2
ment s
o ,
5
ms
a
)
t o
e ,
S
t
a
r
n
i
d
c
( s
Pi
P4
r
P 3.
3
e
, This fig u re does n o t represent an Eu le r graph, f o r there is
p
sno path f ro m p
r
iielement
p(s
e t o
)graph.
3
si
t., s
e
L. O. KATSOFF
o
Boston
College, Ne wt o n (Mass.)
F
n3 u r t h
L. HART
R
Univ.
o
f
Misso
u
ri,
Kansas
C
i
t
y
e
t) r ,
,tw o hu
a
w
REFERENCES
e
tl d
e
iin n o c l
w
[I] ORE, Oy s tein, Th e o ry o f Graphs , A meric an Mat hemat ic al Soc iet y Co lu
s i Public ations , X X X V I I I , Prov idenc e, R. L, 1962, pp. 13.
ot loquium
o
o
ns un
u
o
of f i
lfc e
f
d
sR
t
t
o
iio
l
b
lsm
taa
n
:
a
ok
t
ihe
e