For a graph G=V,E, a subset F of E is called an edge dominating set of G if every edge not in F is adjacent to some edge in F. The edge domination number γ′G of G is the minimum cardinality taken over all edge dominating sets of G. Here, we determine the edge domination number for shadow graphs, middle graphs, and total graphs of paths and cycles.

A dominating set is called a global dominating set if it is a dominating set of a graph G and its complement G¯. A natural question arises: are there any graphs for which it is possible to relate the domination number and the global domination number? We have found an affirmative answer to this question and obtained some graphs having such characteristic.

A set F ⊆ E(G) is an edge dominating set if each edge in E(G) is either in F or is adjacent to an edge in F . An edge dominating set F is called a minimal edge dominating set if no proper subset F of F is an edge dominating set. The edge domination number γ (G) is the minimum cardinality among all minimal edge dominating sets. We investigate the edge domination number of some graphs called snakes which are obtained from path P n by replacing its edges by cycles C 3 and C 4 .

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A dominating set is called a global dominating set if it is a dominating set of a graph and its complement . Here we explore the possibility to relate the domination number of graph and the global domination number of the larger graph obtained from by means of various graph operations. In this paper we consider the following problem: Does the global domination number remain invariant under any graph operations? We present an affirmative answer to this problem and establish several results.