400
IEEE
TRANSACTIONS ON COMPUTERS, VOL. c-33, NO. 5, MAY 1984
Line Digraph Iterations and the
Digraph Problem
MIGUEL A. FIOL, J. LUIS ANDRES YEBRA,
Abstract -This paper studies the behavior of the diameter and
the average distance between vertices of the line digraph of a given
digraph. The results obtained are then applied to the so-called
(d, k) digraph problem, that is, to maximize the number of vertices
in a digraph of maximum out-degree d and diameter k. By line
digraph iterations it is possible to construct digraphs with a number of vertices larger than (d2 - l)/d2 times the (nonattainable)
Moore bound. In particular, this solves the (d, k) digraph problem
for k = 2. Also, the line digraph technique provides us with a
simple local routing algorithm for the corresponding networks.
Index Terms -Communication network, (d, k) graph problem,
line digraph, Moore bound, routing algorithm.
AND
IGNACIO ALEGRE DE MIQUEL
k
Manuscript received November 22, 1982; revised May 31, 1983 and
October 10, 1983.
The authors are with the Department of Mathematics, E.T.S. de Ing. de
Telecomunicacion, Polytechnic University of Barcelona, Spain.
d(x, y).
N2x, YEV
To reduce the number of transit switching stages k and/or
k must be small. In fact, sometimes k cannot exceed.a given
value. In this context, the (d, k) digraph problem consists of
determining the digraph with the largest number of vertices
N for given values of the maximum out-degree d and of the
diameter k. It is easy to see that N is related to d and k by the
Moore bound
N
1 + d + d2 +
...
I. INTRODUCTION
R ECENT advances in VLSI technology and microprocessors have had a great impact on the design of
computer systems, stimulating research on interconnection
networks. These networks can be modeled by graphs whose
N vertices represent the processing elements or nodes of the
network and whose edges represent the links between them.
The graphs thus obtained can be directed or undirected.
We are concerned here with directed graphs only, called
digraphs for short. A digraph D = (V, E) consists of a set V
of points called vertices and a set E of directed edges
between vertices of D. If [x, y] is an edge from x to y, we say
that x is adjacent to y and also that y is adjacent from x. (We
shall assume that D has no parallel edges, that is, that there
is at most one edge from each vertex x to any other vertex y.)
The network must allow communication between any pair
of processing elements. Of course, the complete symmetric
digraph KN in which each vertex is adjacent to all others
satisfies this condition. But for large values of N this solution
is too expensive or even technically unfeasible. So, we are
forced to use digraphs with the property that every vertex is
adjacent to no more than a few, say d, vertices, but such that
there is a (directed) path from any vertex to any other.
The minimum number of edges needed to go from a vertex
x to another vertex y is called the distance d(x, y) from x to y.
(Note that in a digraph d(x,y) is not necessarily equal to
d( y, x).) The maximum distance between any pair of vertices
is the diameter k of the digraph, while the average distance
between vertices of the digraph is defined as
(d, k)
+ d' = NM(d,k)
ik+ 1
=
dk+l
d-
-
if d
1
=
1
if d> 1.
1
(1)'
The Moore bound is attained for d = 1 by a ring digraph in
which every vertex is connected to the following one, and for
k = 1 by the complete symmetric digraph Kd+l. However, it
is known [3] that this bound cannot be attained for d > 1
andk> 1.
Using the results obtained in Section II about the diameter
of a line digraph, we construct in Section III a family of
digraphs that for any values of d > 1 and k > 1 comes close
to the Moore bound. Other constructions are presented in
Section IV.
Another important requirement that should be given
careful consideration is the existence of efficient routing algorithms to find short paths between any two nodes of the
network. In Section V we present a simple local routing -algorithm that finds the shortest path between any two vertices of
the digraph.
IL. LINE DIGRAPH ITERATIONS
In the line digraph L(D) of a digraph D = (V, E), each
vertex represents an edge of D, that is V(L(D)) =
{uv [u, V] E E(D)}; and the vertex uv is adjacent to the vertex
wz iff v = w (i.e., when the edge [u, v] is adjacent to the
edge [w, z] in D). Fig. 1 shows a digraph D and its line
digraph L(D).
The number of vertices in L(D) is
NL
=
IV(L(D))I
=
|E(D)| = veV, d+(v) = E d-(v)
VEV
(2)
where d+(v) and d-(v) are the out- and in-degree of vertex
v, respectively.
0018-9340/84/0500-0400$01.00 (© 1984 IEEE
FIOL et
al.: LINE DIGRAPH ITERATIONS
401
To prove (5), note that for any two vertices i,j E V(D) with
d(i,j) = m there are at most d-(i) * d+(j) = d2 vertices at
distance m + 1 in L(D). To carry out an exact computation
we must leave out from this set one vertex each time that there
is an edge to i that coincides with an edge from j; that is, for
each edge [j, i]. Therefore,
i
\Or,,
L(D) I4
4
kL= NL2
24
40
When D is a d-regular digraph, that is, when d+(v) =
d-(v) = d for all vertices v E V, it has |E| = dN edges. Then
its line digraph L(D) is also d-regular with
(3)
vertices and d2N edges. And for d > 1 the sequence of line
digraph iterations
L(D),L2(D)
=
L(L(D)), * Ln(D)
=
+
1I- E {d(i,j)
[j, i]eE(D)
+
I}) (6)
and so
Fig. 1. A digraph and its line digraph.
NL = dN
d2{d(i,j)
i,jeV(D)
L(Ln-(D)), *
k- < >f(d2
i,jeV(D)
d(i,j) + d2N2) = k +
1.
Remarks: 1) This last bound may be improved upon if the
girth (= length of the minimum cycle) q of D is known. Then
d(i,j) q - 1 whenever [j, i] E E(D) and so
>
e o +a(q
d(i,j)
Er
[j, il]eE(D)
-[j, i]eE(D)
1)
+(
=
qdN
Therefore, we obtain from (6)
is an infinite sequence of d-regular digraphs.
We remark now that each vertex x in L2(D) represents an
kL<k+ 1 -dN
(7)
edge [u, v] of L(D) and that the vertices of this edge corre2) For a nonregular digraph D that is not a cycle, (4) still
spond to two adjacent edges of D, say [io, il] and lil, i2i. So,
the vertex x of L2(D) represents a path of length 2 in D and we holds [5].
may write x = iOili2 because D has no parallel edges. Also,
III. THE (d, k) DIGRAPH PROBLEM
the vertex x will be adjacent to another vertex y iff this vertex
represents an edge of the form [v, w] in L(D). Therefore, y
The theorem in the preceding section shows that, from a
must be equal to ili2i3 with w = [i2, i3] E E(D).
d-regular digraph D with N vertices and diameter k, it is
More generally, each vertex x in Ln(D) represents a path possible to construct a new d-regular digraph L(D) with
* is adjacent to the NL = dN vertices and diameter k + 1. Since from the Moore
ioil ... in of length n in D, and x = iOil in
vertices of the form y = j1j2 * i with [in, in+i] EE (D). bound we have
As we are going to see, this will allow us to implement
dk+2~ dk+2dd
d
routing algorithms that are local and very simple.
+
1 dNm(d, k) +
The following result illustrates the interest of the line Nm(dd k 1)+=
digraph concept in order to obtain directed graphs with a
the above-mentioned result is close to the optimum, and
large number of vertices and small values of k and k.
Theorem: Let D = (V, E) be a d-regular digraph (d > 1) therefore it must have far-reaching consequences in conof order N, diameter k, and average distance between vertices nection with the (d, k) digraph problem.
To begin with, the maximum number of verticesf(d, k) in
k. Then, the order NL, diameter kL, and average distance
a
d-regular
digraph of diameter k must satisfy
between vertices kL of L(D) satisfy
N~dk+1~
1
f(d,k + 1) df(d,k)
(8)
(3)
(4) because the line digraph L(D) of a d-regular digraph with
f(d, k) vertices already has df(d, k) vertices.
(5)
In particular, from f(d, 1) = d + 1 we obtain
Proof: (3) has already been deduced from (2). To prove
d(d + 1) S f(d, 2) < NM(d, 2) = d2 + d + 1
NL
:
dN
kL= k + 1
kL< k + 1.
=
(4), note that to go from the vertex u = ij to the vertex
v = pq, v + u, in L(D) is equivalent to going from the edge
[i, j] to the edge [p, q ] in D. But the path of minimum length
with these two terminal edges contains the shortest path between j and p. Therefore,
dL(D)(u, V) = dD(j,p) + 1
This will imply kL = k + 1 if there are at least two different
edges [i, j], p, q ] E E(D) such that d(j, p) = k. But, as
d-(j) = d 2, at least one of the d edges [.,j] must be
different from [p, q].
,
so that
f(d, 2)
=
d2 + d
(9)
and this value is attained by L(Kd+I).
Also, the digraphs Dkd = Lkl- (Kd+l) are d-regular, with
diameter k and number of vertices
d2 lk+I1
dk+ld2~
d-1
N =dl(d + 1) = d2 d-l d>
=
d2
1
d2NM(d, k)
(10)
402
IEEE TRANSACTIONS ON COMPUTERS, VOL.
c-33,
NO.
5, MAY 1984
D2
02
Fig. 3. A 2-regular digraph of diameter 4 on 25 vertices.
Fig. 2. The digraphs Dl = K3,D2 = L(K3) and D2 = L2(K3).
that is, N is larger than (d2 - 1)/d2 times the (nonattainable)
Moore bound. This family of digraphs was already described
in [9] as shift register state diagrams. Other constructions can
be found in [6] and [11]. We show in Fig. 2 the first three
members of the family corresponding to d = 2.
We expect that f(d, k) must not be too far from N =
(d + 1)dk . We should point out, however, that for d = 2,
f(2, k) 25 * 2k4 because the graph of Fig. 3 (found by
computer search) has 25 vertices and diameter 4.
Finally, we mention that, for solutions of the (d, k) digraph
problem, all vertices must have the same out-degree since
otherwise there would be a vertex with out degree d+(v) >
d - 1, and this implies
-
N
-<
I + (d
-
1) + (d-l1)d + *
+
whereas the digraphs Lkl-(Kd+I) have N =
(d
G2
G2
2
L (G2)
Fig. 4. The digraph G2 and its line digraph G3.
I)d k-I = d k
dk + d`k vertices.
-
TABLE I
Dk
IV. OTHER CONSTRUCTIONS
It is easy to see that the line digraph technique transforms.
a cycle of a digraph D in a cycle of the same length in L(D).
In particular, L(D) has the same girth as D. As Kd±l has (d21)
digons (= cycles of length 2), L`- (Kd+l) also has (d21)
digons, and thus its girth is equal to 2.
To improve the average distance between vertices it is
better to start out with digraphs that have no digons. For
example, for d = 2 it is possible to construct another family
of digraphs with the same values of N for each k : 2,
N = 2k + 2k-1, but with a smaller average distance between
vertices. This is the family of line digraph iterations of the
digraph G2 of Fig. 4: G2 = Lk-2(G2) for k ¢ 2. In Table I we
compare for k 7 the values of k and those given by (7) for
the two families of graphs D2 and GkWhen the number of vertices is of the form N = dm the line
digraph method can be used to obtain the digraphs proposed
in [7] and [9]. In fact, the digraph with dk vertices and
diameter k, that we call Fk, can be viewed as Lk-i(F1) where
Fd is the d-regular digraph with d vertices, in which every
vertex is adjacent to all vertices including itself. Fig. 5 shows
the first three members of the family corresponding to d = 2.
This family of digraphs was already known in a different
context (see, for example, [4]).
For our purposes we are not interested in the presence of
self-loops because these are useless links. Leaving out the d
self-loops of a digraph Fd and connecting the affected vertices to and from a new vertex z we get a new d-regular digraph
=
k
N
1
2
3
4
5
6
7
3
6
12
24
48
96
192
0.6667
1.3333
2.1250
2.9896
3.9037
4.8516
5.8205
G2
bound (7)
k
bound (7)
1.3333
2.1667
3.0417
3.9479
4.8828
5.8411
1.3333
2.0833
2.9427
3.8568
4.8050
5.7744
2.0833
2.9583
3.8802
4.8256
5.7894
01
F2F2
F1
-
F2F3
110
.100
Fig. 5. The digraphs Fl, F2
=
L(Ff), and F3
=
L2(Fl).
of diameter m and number of vertices dm + 1 (see Fig. 6).
The diameter is still m because every vertex x of Fd is at
distance -m - 1 from one of the modified vertices, so at a
distance from z of d(z, x) m, and analogously d(x, z) m.
Now, by line digraph iterations we can obtain d-regular digraphs of diameter k and number of vertices dk-m(dm + 1) =
dk + dkm for 0 < m < k (see also [8]).
Finally, leaving out the orientations of the edges in
-
-
x
FIOL et al.: LINE DIGRAPH ITERATIONS
403
REFERENCES
F3
F3
+{z}
12
Fig. 6. The digraphs F3 and F3 +
{z}.
Lk-'(Kd+l),
a
=
we get undirected regular graphs of degree
2d, diameter k, and number of vertices
(A)k
improving
some
+()k-l
of the best known values given in [2].
V. ROUTING
Let x = xox. x, and y = YoYI .Y. be two vertices of
L"(D). We are looking for a short path from x to y. As we
have seen, each of vertices x and y may be interpreted as a
path of length n in D. Suppose that the shortest path x, = zo,
zl,
zp = yo from xn to yo in D is known. (This is trivial if
D is a complete symmetric digraph.)
In the sequence
,
Xr
Xo, XI,
=
Z0 Z1,
successive points form an edge of D and any set of
n + 1 successive points represents a vertex of L"(D). Also,
any set of n + 2 successive points represents an edge of
Ln(D), and two successive such sequences have a common
(n + 1)-sequence, that is, a vertex of Ln(D). This leads to a
path from x to y of length
p+n k+n
where k is the diameter of D.
If for some a
,a
(11)
for i = 0, 1,
,
Xn-a+i =Yi
the remarks above apply also to the sequence
any two
vXn,- = YO,
in this way with
XO, XI,
providing
x
to y.
X
=
us
,
a
Xn
=
Yn
Ya,
path of length
n
-
XnZI
Evidently, d(x',y)
and continue.
=
d(x,y)
-
1,
so
3)
go
communication and Dr.
Engineer degrees
respectively, both from
nica Superior de Ingenieros de
and 1982,
the
Polytechnic University
in 1979
the Escuela
Tec-
Telecomunicaci6n
at
of Barcelona.
In 1979, he joined the Department of Mathematics
of the E.T.S. de Ing. de Telecomunicaci6n, where he
is now an Assistant Professor. His current research interest is the applications
of graph theory to computer architecture, especially the study of interconnection topologies for large computer networks. Presently, he is involved
in a research project on the design and evaluation of interconnection networks
for multiprocessor systems.
J. Luis Andres Yebra was born in Le6n, Spain, on
May 21, 1945. He received the B.Sc. degrees in both
mathematics and telecommunications
engineering
from the University of Madrid in 1968, where
he also obtained the Ph.D. degree in mathematics
in 1975.
a
from
These considerations can be used to produce a local routing
algorithm that uses the shortest path between x and y in
the following way. 1) If a is the greatest integer such
that (11) holds set x' = X1X2 .XnYa+I. 2) Otherwise, if
xnzI... zp-lyO is the shortest path from x. to yo in D, set
X1X2
Miguel A. Fiol was born in Palma de Mallorca,
Spain, on December 20, 1949. He received the
Technical Engineer in Electricity degree from
the Escuela de Ingenieria Tecnica Industrial de
Barcelona in 1974, and the Engineer of Tele-
Yn
Yo,YI,
I, Zp
[1] 1. Alegre, J. L. A. Yebra, M. A. Fiol, M. Valero, and T. Lang, "An
algorithm to minimize the diameter of directed graphs," 2nd Spanish
Symp. on Distributed Systems, Santiago de Compostela, Spain,
Sept. 1982.
[2] J. C. Bermond, C. Delorme, and J. I. Quisquater, "Tables of large graphs
with given degree and diameter," Inform. Process. Lett., vol. 15,
pp. 10-13, 1982.
[3] W. G. Bridges and S. Toueg, "On the impossibility of directed Moore
graphs," J. Comb. Theory B, vol. 29, pp. 339-341, 1980.
[4] N. Deo, Graph Theory with Applications to Engineering and Computer
Science. Englewood Cliffs, NJ: Prentice-Hall, 1974.
[5] M. A. Fiol, "Applications of graph theory to interconnection networks"
(in Spanish), Ph.D. dissertation, Polytech. Univ. of Barcelona, Spain,
1982.
[6] M. A. Fiol, Alegre, and J. L. A. Yebra, "Line digraph iterations and the
(d, k) problem for directed graphs," in Proc. 10th Int. Symp. Comput.
Arch., 1983, pp. 174-177.
[7] M. Imase and M. Itoh, "Design to minimize diameter on building-block
networks," IEEE Trans. Comput., vol. C-30, pp. 439-442, June 1981.
[8] -, "A design method for directed graphs with minimum diameter,"
IEEE Trans. Comput., vol. C-32, pp. 782-784, Aug. 1983.
[9] W. H. Kautz, "Design of optimal interconnection networks for rpultiprocessors," in Architecture and Design of Digital Computers. NATO
Advanced Summer Institute, 1969, pp. 249-272.
[10] G. Memmi and Y. Raillard, "Some new results about the (d, k) graph
problem," IEEE Trans. Comput., vol. C-31, pp. 784-791, Aug. 1982.
[11] S.M. Reddy, D.K. Pradhan, and J.G. Kuhl, "Directed graphs with
minimum diameter and maximum node connectivity," School of Eng.,
Oakland Univ., Rochester, MI, Tech. Rep., July 1980.
[12] S. Toueg and K. Steiglitz, "The design of small diameter networks
by local search," IEEE Trans. Comput., vol. C-28, pp. 537-542,
July 1979.
from x to x'
ACKNOWLEDGMENT
The authors sincerely acknowledge the valuable
gestions and corrections of the referees.
sug-
Since 1976 he has beep Professor in the Departof Mathematics of the E.T.S. de Ing. de
Telecomunicaci6n at the Polytechnic University of
Barcelona. His current research area is the study of
interconnection networks by graph theory methods.
Presently, he is engaged in a research project on the design and evaluation of
interconnection networks for multiprocessor systems.
ment
Ignacio Alegre de Miquel was born in Barcelona,
Spain, on July 31, 1953. He received the B .S. degree
in mathematics from the University of Barcelona
in 1975.
From 1977 to 1979 he was a Teaching Assistant
in the Department of Numerical Analysis at the
University of Barcelona. Since 1980 he has been a
Teaching and Research Assistant in the Department
of Mathematics at the Polytechnic University of
Barcelona.