AbstractIn this paper, we describe the structure of separable self-complementary graphs

Kawarabayashi, Ken-ichi

Nakamoto, Atsuhiro

Oda, Yoshiaki

Ota, Katsuhiro

Tazawa, Shinsei

Watanabe, Mamoru

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On separable self-complementary graphs 

Discrete Mathematics 257 (2002) 165–168
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On separable self-complementary graphs
Ken-ichi Kawarabayashia , Atsuhiro Nakamotob;∗, Yoshiaki Odac ,
Katsuhiro Otaa , Shinsei Tazawad , Mamoru Watanabee
aDepartment of Mathematics, Keio University, Yokohama 223-8522, Japan
bDepartment of Mathematics, Osaka Kyoiku University, 4-698-1 Asahigaoka, Kashiwara,
Osaka 852-8582, Japan
cDepartment of Mathematics and Computer Science, Shimane University, Matsue 690-8504, Japan
dDepartment of Mathematics, Kinki University, Higashi-Osaka, 577-8502, Japan
eDepartment of Computer Science and Mathematics, Kurashiki University of Science and the Arts,
Kurashiki 712-8505, Japan
Received 23 April 2001; received in revised form 15 September 2001; accepted 8 October 2001
Abstract
In this paper, we describe the structure of separable self-complementary graphs. c© 2002
Elsevier Science B.V. All rights reserved.
Keywords: Self-complementary graph; Cut vertex; Separable graph
1. Introduction
We consider only simple 9nite undirected graphs. We denote the vertex set and edge
set of a graph G by V (G) and E(G), respectively. A graph G is said to be trivial if
G has just one vertex, and G is empty if G has no vertex. A graph G is said to be
separable if G has a cut vertex. For a graph G, the complement of G is the graph
denoted by <G such that V ( <G)=V (G) and E( <G)= {uv: uv ∈E(G)}.
The following is well known and easy to see:
For every graph G; G itself or its complement <G is connected:
We extend this observation to 2-connected graphs. That is, we would like to consider
whether or not, for every graph G; G or <G is 2-connected. If not, we want to char-
∗ Corresponding author. Present address: Department of Mathematics, Faculty of Education and Human
Sciences, Yokohama National University, 79-2 Tokiwadai, Hodogaya-Ku, Yokohama 240-8501, Japan.
E-mail address: nakamoto@edhs.ynu.ac.jp (A. Nakamoto).
0012-365X/02/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.
PII: S 0012 -365X(02)00399 -0
166 K.-i. Kawarabayashi et al. / Discrete Mathematics 257 (2002) 165–168
acterize such a graph. Note that by the above statement, if both G and <G are not
2-connected, then at least one of G and <G has a cut vertex. We 9rst establish the
following theorem.
Theorem 1. Let G be a separable graph with a cut vertex v. Then the complement
<G of G is separable or disconnected if and only if (a) or (b) hold:
(a) degG(v)¿|V (G)| − 2,
(b) G has a vertex u of degree 1 such that G− u has a spanning complete bipartite
subgraph.
A graph G is said to be self-complementary if G and <G are isomorphic. Let G
be a self-complementary graph with n vertices. For which n¿2 are there self-comple-
mentary graphs with precisely n vertices? Since |E(G)|= |E( <G)| and |E(G)|+|E( <G)|=
n(n− 1)=2, the number n(n− 1)=2 must be even, and hence we have n≡ 0; 1 (mod 4)
[1]. Moreover, for every natural number n≡ 0; 1 (mod 4) with n¿4, we can construct
self-complementary graphs, as follows.
Let H be a graph which is either empty, trivial, or self-complementary, and let
P4 = v1v2v3v4 be a 4-path, i.e., a path with exactly four vertices. Join each of v2 and v3
to all vertices of H . We call this operation a 4-path addition. The resulting graph with
|V (H)| + 4 vertices can easily be checked to be self-complementary. Thus, for each
n≡ 0; 1 (mod 4) with n¿4, we can inductively construct self-complementary graphs
with precisely n vertices.
However, the self-complementary graphs obtained as above always have cut vertices
and hence are separable. Our second result claims that the converse is also true.
Theorem 2. Let G be a separable self-complementary graph with at least 4 vertices.
Then, G can be obtained from a graph H by a 4-path addition, where H is either
empty, trivial or self-complementary.
Let H1 and H2 be two self-complementary graphs and let G1 and G2 be the two
self-complementary graphs obtained from H1 and H2 by a 4-path addition, respectvely.
Then, it is easy to see that H1 and H2 are isomorphic if and only if so are G1 and
G2. Therefore, we can obtain the following corollary from Theorem 2. (Note that we
do not regard a trivial graph as a self-complementary graph.)
Corollary 3. The number of the separable self-complementary graphs with n¿8
vertices is the same as the number of the self-complementary graphs with n − 4
vertices.
The number of the self-complementary graphs with n vertices has already been
determined in [3]. Thus, by Corollary 3, we can obtain Table 1.
In general, we can construct many nonseparable self-complementary graphs by a
method similar to a 4-path addition, as follows. Let B be a graph with k¿2 vertices and
K.-i. Kawarabayashi et al. / Discrete Mathematics 257 (2002) 165–168 167
Table 1
The number of separable self-complementary graphs with n vertices
The number of vertices 4 5 8 9 12 13 16 17
Self-comp. 1 2 10 36 720 5600 703 760 11 220 000
Separable self-comp. 1 1 1 2 10 36 720 5600
Nonseparable self-comp. 0 1 9 34 710 5564 703 040 11 214 400
let H be a self-complementary graph. Let B1 =B4 =B and B2 =B3 = <B. Join all vertices
of Bi and all vertices of Bi+1 for i=1; 2; 3, and join all vertices of Bj and all vertices
of H for j=2; 3. Then the resulting graph is k-connected and self-complementary.
2. Proof of theorems
Proof of Theorem 1. The suMciency is obvious and hence we prove the necessity. Let
v be a cut vertex of G and let B1; : : : ; Bk be the components of G − v. Then G − v
has a spanning complete k-partite subgraph. Here, if degG(v)¿|V (G)| − 2, then v has
degree at most 1 in <G. This is case (a).
We consider other cases. Since we may assume deg <G(v)¿2 in this case, <G is
2-connected if and only if G − v is 2-connected. Thus, we check the 2-connectivity of
G − v. If k¿3, then G − v includes a spanning complete k-partite subgraph and hence
G − v is 2-connected. Thus, we have k =2. If |V (B1)|¿2 and |V (B2)|¿2, then G − v
has K|V (B1)|;|V (B2)| as a spanning subgraph, and hence it is 2-connected.
Thus, we may suppose that |B1|=1 and put B1={u}. Since deg <G(v)¿2 and since u
is adjacent to all vertices x∈V (G)− {u; v} in <G; <G is not 2-connected if and only if
G − u is not connected. Clearly, G − u is not connected if and only if G − u has a
spanning complete bipartite subgraph. This is case (b).
Proof of Theorem 2. Let G be a separable self-complementary graph with n¿4 ver-
tices. Then, by Theorem 1, G has a vertex u of degree 1. Since G is self-complementary,
G has a vertex u′ of degree n− 2. Let ’ : G→ <G be an isomorphism.
We 9rst claim that G has a vertex v of degree 1 other than u. If u is a unique
vertex of degree 1 in G, then ’(u)= u′ and ’(u′)= u. If u and u′ are adjacent in one
of G and <G, say G, then ’(u)(= u′) and ’(u′)(= u) are not adjacent in <G, which
contradicts that ’ is an isomorphism. Thus, G has at least two distinct vertices u and v
of degree 1, and hence G also have distinct vertices u′ =’(u) and v′ =’(v) of degree
n− 2 in G.
Let u1 and v1 be the unique neighbors of u and v in G, respectively. Let
A=V (G) − {u; v; u1; v1}. If A= ∅, then V (G)= {u; v; u1; v1}, and hence G itself is a
4-path with {u1; v1}= {u′; v′}. Therefore, the theorem holds when n=4.
So we may suppose that A = ∅. Observe that each vertex x∈A can have at most
n − 3 neighbors in G since u and v have degree 1 in G. Thus, u1 and v1 must be
distinct and {u1; v1}= {u′; v′}. Without loss of generality, we may put u1 = u′ and
168 K.-i. Kawarabayashi et al. / Discrete Mathematics 257 (2002) 165–168
v1 = v′. Moreover, u1 is adjacent to all vertices of G except v, and v1 is adjacent to
all vertices in G except u.
Now we can 9nd the 4-path uu′v′v such that degG(u)= degG(v)= 1 and degG(u
′)=
degG(v
′)=n− 2, and other vertices x∈A have degree 26degG(x)6n− 3. So, ’ maps
{u; u′; v′; v} to {u; u′; v′; v}. Since the 4-path induced by {u; u′v′; v} is self-complemen-
tary and since each of u′ and v′ is adjacent to all vertices in A, the subgraph 〈A〉
induced by A in G must be either trivial or self-complementary. Thus, the theorem
holds for n¿5.
References
[1] F. Harary, E.M. Palmer, Enumeration of locally restricted digraphs, Canad. J. Math. 18 (1966) 853–860.
[2] K.R. Parthasarathy, Sridharan, Enumeration of self-complementary graphs and digraphs, J. Math. Phys.
Soc. Madras 3 (1969) 410–414.
[3] R.C. Read, On the number of self-complementary graphs and digraphs, J. London Math. Soc. 38 (1963)
99–104.


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On separable self-complementary graphs

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