### Abstract:

For a given graph G = (V, E), a dominating
set D ⊆ V (G) is said to be an outer connected dominating set if
D = V (G) or G − D is connected. The outer connected domination
number of a graph G, denoted by γec(G), is the cardinality of a
minimum outer connected dominating set of G. A set S ⊆ V (G) is
said to be a global outer connected dominating set of a graph G if S
is an outer connected dominating set of G and G. The global outer
connected domination number of a graph G, denoted by γegc(G), is
the cardinality of a minimum global outer connected dominating set
of G. In this paper we obtain some bounds for outer connected dom-
ination numbers and global outer connected domination numbers
of graphs. In particular, we show that for connected graph G =6 K1,
max{n −
m+1
2
,
5n+2m−n
2−2
4
} 6 γegc(G) 6 min{m(G), m(G)}. Fi-
nally, under the conditions, we show the equality of global outer
connected domination numbers and outer connected domination
numbers for family of trees.