Hegde, N.

Read, R.C.

Sridharan, M.R.

Elsevier - Publisher Connector 

Discrete Mathematics 47 (1983) 109-112 
North-Holland 
109 
NOTE 
THE ENUMERATION OF TRANSITIVF, SEXF- 
COMPLElMENT ARY DIGRAPHS 
N. HEGDE, R.C. READ and M.R. SRIDHARAN” 
Department of Combinatorics, University of Waterloo, Waterloo, Ontario, Canada N2L 3Gl 
Received 3 February 1981 
Revised 2 November 1982 
1. introduction 
In this paper we obtain a formula for the number of transitive self- 
complementary digraphs on any given number of nodes. The numbers of these 
digraphs were found by one of us, in [l], by a completely different method. Here 
we give a shorter proof of this result - one that also shows how to construct all 
digraphs of this kind. 
2. Texminology 
If (a, b) is a (directed) arc in a digraph D we say that a is ‘adjacent to’ b, or that 
6 is ‘adjacent from’ a. If the arcs (a, b) and (b, a) are both present in D we say, 
for brevity, that a is ‘adjacent with’ b or that a and b are ‘adjacent both ways’. If 
neither (a, b) nor (b, a) occur, a and b are ‘non-adjacent’. 
A digraph C is transitive if the existence of arcs (a, 6) and (b, c) always implies 
the existence of the arc (a, c). D is self-complementary if it is isomorphic to its 
complement. 
A ‘clique’ in D is a set C of nodes such that any two nodes in C are adjacent 
both ways, and C is maximal for this property. A ‘maximal independent set’ in D 
is a set I of nodes no two of which are adjacent either way, and which is maximal 
for this property. A clique of size 1 is indistinguishable from a maximal indepen- 
dent set of size 1; we shall call such sets, consisting of a single node, ‘singletons’. 
In much of what follows it will not be important whether a set forms a clique or a 
maximal independent set, and for this reason we introduce, temporarily, a generic 
term embracing both these concepts. We define an ‘M-set’ in D to be a set of 
nodes that is either. 
* Permanent address: Indian Institute of Technology, Kanpur-208016, India. 
0012-365X/83/$3.00 0 1983, Elsevier Science Publishers B.V. (North-Holland) 
110 N. Hegde, et al. 
(a) a clique of size 32, 
(b) a maximal independent set of size 22, or 
(c) a singleton. 
3. preliminary resuha on the stmeture of transitive self-complementary digraphs 
One obvious property of transitive self-complementary digraphs is that under 
complementation cliques map into maximal independent sets and vice versa, while 
singletons map into singletons. 
Let A and B be any two distinct M-sets in a transitive self-complementary 
digraph D. 
Theorem 1. A and B are disjoint. 
Proof. This is obvious if either A or B is a singleton; and there is an immediate 
contradiction if we suppose A and B to have more than one node in common. Let 
x E A n B, a E A and b E B. If A and B are both cliques then, by transitivity, a is 
adjacent to b, which is not so. If A and B are both maximal independent sets we 
get a similar contradiction by considering the complement D of D. Suppose A is a 
maximal independent set and B is a clique. Then a is not adjacent to b since, if it 
were, it would be adjacent, by transitivity, to x. Similarly a is not adjacent from b. 
Hence, in Is, a is adjacent with b as well as with x. It follows, by transitivity, that 
b is adjacent with x in D, which contradicts the fact that these nodes.are adjacent 
in D. 
Theore!m 2. Either every node of A is adjacent to every node of B, or every node of 
B is adjacent to every node of A. 
Proof. If a (in A) and b (in B) are adjacent with each other they both belong to 
some clique. This contradicts Theorem 1. By considering D we see that a and b 
cannot be non-adjacent. Hence between any node of A and any node of B there 
is an arc in just one direction. 
Without loss of generality (considering D if necessary) we may assume that one 
of the M-sets, say B, is a clique. Suppose there are arcs (aI, b,) and (b,, a,) with 
al, a2 E A and bl, b2 E B. Then, by transitivity, al is adjacent to a2, so that A must 
be a clique. And if so then it is easily seen that a2 is adjacent with b2, which we 
have already shown to be impossible. This complets the proof of Theorem 2. 
From Theorem 1 and Theorem 2 it follows that the M-sets of D form a 
partition of the nodes of D, and that the arcs between any two M-sets are all in 
the same direction and all possible edges in that direction occur. 
Let us now shrink each M-set in D to a single node, and replace all the edges 
between two M-sets by a single edge. The resulting digraph must be a tournament 
The enumeration of transitive self -complementary digraphs 111 
(by Theorem 2) and must be acyclic, since the nodes of any cycle would, by 
transitivity, be mutually adjacent. But an acyclic tournament is simply a linear 
ordering, i.e. a digraph whose nodes can be numbered 1,2, . . . , n so that node i is 
adjacent to node i if and only if i < j. 
Hence D consists of a number of M-sets arranged in a ‘chain’ as shown in Fig. 
1, with each M-set adjacent (in the sense of Theorem 2) to each M-set to its right. 
Moreover, since complementation reverses all the arrows and interchanges cliques 
and maximal independent sets, the sizes of the M-sets (m,, m2, . . .) must read the 
same from each end of the chain; and a clique on one side of the chain will be 
matched by a maximal independent set in the corresponding position on the other 
side of the chain. 
If there is a central M-set it must be a singleton, and D then has an odd number 
of nodes. Removing this singleton from the chain gives a unique digraph on one 
fewer nodes, and conversely. From this we deduce 
Theorem 3. The number of transitive self-complementary 
is the same as the number on 2n nodes. 
digraphs on 2n + 1 nodes 
4. The enumeration 
We now have a method cf constructing transitive self-complementary digraphs 
on any even number, 2n, of nodes (and hence, by Theorem 3, on any odd number 
of nodes). We choose k positive integers ml, m2, . . . , mk whose sum is n, and 
arrange M-sets of these sizes as in Fig. 1. When m, > 1 we also have to choose 
whether the ith M-set on the left-hand side of the chain is a clique or a maximal 
independent set (the corresponding M-set on the other side is then the other 
choice). The chain of Fig. 1 is then determined and hence so is D. By the results 
of Section 3, every transitive self-complementary digraph can be obtained in this 
way, and different choices of k, the m,, and of clique or maximal independent set 
will give different digraphs. 
For a given value of k the number of choices is 
C f !mbf (m2) - * - f-b-k) 
where the summation is for all {mi} such that 1 mi = n, and where f (1) = 1 and 
Fig. 1. 
112 N. Hegde, et al. 
f(m)=2 if m *2. This summation is easily seen to be the coefficient of X” in 
(x + 2x2 + 2x3+2x4 + - - -)k, i.e. in 
Hence the total number of transitve self-complementary digraphs is the coeffi- 
cient of xn in 
x+x2 k 
f 4xn=kg0 l_x 
l-x 
n=O ( ) =1-2x_x2. 
To obtain a recurrence formula for the numbers a, we multiply the above 
equation by 1-2x -x2 and equate coefficients of xX. We obtain the following 
result. 
Theorem 4. The number a, of transitive self-complementary digraphs on 2n nodes 
is given by the recurrence 
a, = 244 + an-2 
together with the initial conditions a, = al = 1. 
The following explicit formula for a, is readily deduced: 
a, = ${( 1 + J2)n + (1 - J2)“). 
We are grateful to the referee for pointing out that this problem can be 
interpreted and solved very elegantly using the language and known properties of 
partially ordered sets. The proof given here is perhaps more in keeping with the 
original graph-theoretical formulation of the problem. 
[l] Malati Hegde, Some enumerative digraph and hypergraph problems, Ph.D. Thesis, 1.1-T. Kanpur, 
India, August 1978. 


The enumeration of transitive self-complementary digraphs 

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The enumeration of transitive self-complementary digraphs

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