On edge domination numbers of graphs
Baogen Xu 1
Dept. of Mathematics, East China Jiaotong University, Nanchang 330013,P.R.China
Abstract: In recent years, some kinds of domination in graphs have been investigated[1~2].
Most of those belong to the vertex domination of graphs, A few of results have been obtained
about the edge domination of graphs. In [3~6] we introduce three kinds of edge domination of
graphs, which are called signed edge domination、signed star domination and signed clique
domination, resp. In this paper we state some results for three kinds of edge domination of graphs,
and pose several open problems and conjectures.
1. Introduction
Let G= (V,E) be a graph, for e = uv E(G), N G (e) =｛ e  E(G)| e is adjacent to e｝is called
the open edge- neighborhood of e in G, and N G [e] =N G (e)∪｛e｝is called the closed one. If v  V,
then E G (v) ={uv  E∣u  V } is called the edge- neighborhood of v in G. For simplicity,
N G [e] and E G (v) are denoted by N[e] and E(v), respectively.
A clique K of G is a maximal complete subgraph of G. That is, any complete subgraph (except
for K )of G contains no K as its subgraph. The vertices number of a maximum clique of G,
denoted by  (G ), is called the clique number of G.
Definition 1[3]. Let G= ( V(G), E(G) ) be a graph. A function
f ：E(G) → ｛+1,－1｝is called the
 f (e) ≥1 for every e  E(G). The signed
signed edge domination function (SEDF) of G if
eN [ e ]
edge domination number  s（G）of G is defined as  s（G）= min｛
 f (e) ∣f is an SEDF
eE ( G )
of G｝. For any totally disconnected graph G = K n , then define  s （G）= 0.
Definition 2[4~5] Let G = (V, E ) be a graph without isolated vertices. A function f : E  {+1
–1}is called the signed star domination function ( SSDF ) of G if
 f (e) ≥ 1
for every
eE ( v )
v  V (G ) .The signed star domination number of G is defined as  ss (G) = min {  f (e) ∣ f is an
eE
SSDF of G }. We define  s （G）= 0 for all totally disconnected graphs G = K n .
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Definition 3[6] Let G = ( V, E ) be a graph, a function f : V   1,1 is said to be a signed
clique domination function (SCDF) of G if
 f (v)  1holds for every clique K of G. The
vV ( K )
signed clique domination number

 sc (G ) of G is defined as  sc (G ) = min  f (v) f is an
vV
SCDF of G } . We define  s c （G）= 0 for all totally disconnected graphs G = K n .
By Definition 1~3 we have
Lemma 3 For any two vertex- disjoint graphs G1 and G2 , we have
 s (G1  G2 ) =  s (G1 ) +  s (G2 ) ,
 ss (G1  G2 )   ss (G1 )   ss (G2 ) and
 sc (G1  G2 )   sc (G1 )   sc (G2 )
Obviously, a signed star domination function is a signed edge domination function of G, thus we
have
 (G)≥  s （G）.
Lemma 4 For any graph G without isolated vertices,  ss
2. Main results
Theorem A[3] Let (m)  min  s (G) G
is a graph of
size m and m  1 ,
 1  24m  25  6m  5  
then  (m)  2 
  m .
6
3 

Theorem B[3] Let G be a connected graph, then  s (G)  E (G) if and only if either
G  Pn for some n(1  n  5) or G is the subdivison of some star K 1,n .
Theorem C[4]
Let G be a graph with  (G ) ≥1, then  s （G）≥ V (G)  E(G)
and this bound is sharp.
Statement : Given a graph H with  (H ) ≥1, then there exists a graph G such that H≤G and
 s （G）= V (G)  E(G)
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let G be the graph obtained from H by adding d H (u)  1 pendant edges in vertex u for every
u  V (H ) , define an SEDF f of G as follows:
When e  E ( H ), f (e)  1; when e  E (G ) \ E ( H ), f (e)  1 . We get a signed graph
(G, f ) , so  s （G）≤
 f (e) | E ( H ) |   [d (u)  1] = | V ( H ) |  | E ( H ) | =
eE ( G )
uV ( H )
V (G)  E(G) . By (2), we have  s （G）= V (G)  E(G)
Theorem D[4]
 (G)≤2n – 4,
For any graph G of order n (n≥4), then  ss
and this bound is sharp.
11

n 1 .
6

Theorem E[5]
For any graph G of order n,  s (G ) ≤ 
Theorem F[6]
For any graph G of order n, if    (G )  1, then
 2n 
n
   1
 sc (G )  2 
Theorem G[6] For any connected graph G of order n  2 , if  (G )  s , then
 sc (G ) ≥ 1 
( s 2  n 2 )( s  1)
2s
3. Some open problems and conjectures
 (G ) .
Problem 1 Characterize the graphs which satisfy the equality  s (G ) =  ss
Although in [3] we have determined the exact value of  (m ) = min{  s (G ) |G is a graph of
size m} for all positive integers m, it seems more difficult to solve the following
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Problem 2 [3 ] Determine the exact value of g(n)= min{  s (G ) |G is a graph of order n} for every
positive integer n.
Conjecture 3 [4~5] For any graph G of order n (n≥1),  s (G ) ≤ n  1
If true, the super bound is the best possible for odd n .For example, let G be the subdivision of
the star K
1,
n 1
2
, clearly,  s (G ) =|E(G)| = n-1( the subdivision of a graph G is the graph obtained
from G by subdividing each edge of G exactly once).
Conjecture 4[6] For any connected graph G of order n  2 ,
 sc (G)  1 
(n  1) 2
3
Conjecture 5[6] For any graph G of order n , then
n2 
.
4
 sc (G)  
If true, this upper bound is the best possible. For example, this equality can hold for complete
n
n
bipartite graph G  K s ,t , where s    and t    .
2
2
References
[1] T.W.Haynes, S.T.Hedetniemi, and P.J.Slater, Fundamentals of domination in graphs[M],
New York,1998.
[2]T.W.Haynes,S. T.Hedetniemi, and P.J.Slater, Domination in graphs[M], New York,1998.
[3] Baogen.Xu ,On signed edge domination numbers of graphs[J] . Discrete Mathematics, 239 (2001) 179～189
[4 ]Baogen. Xu ,On edge domination numbers of graphs[J]. Discrete Mathematics,294 (2005) 311～316
[5]Baogen. Xu ,Two classes of edge domination in graphs[J] Discrete Applied Mathematics,154 (2006) 1541～
1546
[6] Baogen. Xu ,Signed clique domination in graphs[J] International Math. Journal. (to appear)
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