Global outer connected domination number of a graph

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.