^{1}

^{2}

^{1}

^{2}

For a graph

The domination in graphs is one of the concepts in graph theory which has attracted many researchers to work on it because of its many and varied applications in such fields as linear algebra and optimization, design and analysis of communication networks, and social sciences and military surveillance. Many variants of dominating models are available in the existing literature. For a comprehensive bibliography of papers on the concept of domination, readers are referred to Hedetniemi and Laskar [

We begin with simple, finite, connected, and undirected graph

The minimum cardinality of a dominating set of

The open neighborhood

For any real number

An edge

In a graph

A subset

Throughout the paper,

We will give brief summary of definitions which are useful for the present investigations.

The open neighborhood of an edge

The degree of an edge

The line graph of

The shadow graph of a connected graph

The middle graph of a connected graph

they are adjacent edges of

one is a vertex of

The total graph of

they are adjacent edges of

one is a vertex of

they are adjacent vertices of

It is easy to see that

For the various graph theoretic notations and terminology, we follow West [

Generally, the following types of problems are considered in the field of domination in graphs:

to introduce new types of dominating models,

to determine bounds in terms of various graph parameters,

to obtain the exact domination number for some graphs or graph families,

to study the algorithmic and complexity results for particular dominating parameters, and

to characterize the graphs with certain dominating parameters.

The present work is intended to discuss the problem of the third kind in the context of edge domination in graphs. In this paper, we investigate the edge domination number of middle graphs, total graphs, and shadow graphs of

Consider two copies of

For

For

Since each edge in

Moreover, the above set

Hence, the above set

Thus,

Let

Then

Now, the edge sets

For

Since each edge in

Now, each graph

Hence, the above set

Thus,

For path

Let

Now, we construct the edge sets of

The above set

Now, the sets

Hence, the above set

This implies that

For cycle

Consider two copies of

First, we construct an edge set of

Since each edge in

Now, the above set is an MEDS of

Hence, the above set

Thus,

Let

Now, we construct an edge set of

Since each edge in

Now,

Hence, the above set

Thus,

For cycle

Let

First, we construct an edge set of

The above set

Now,

Hence, the above set

Thus,

Here, we have taken up a problem to determine the edge domination number for the larger graphs obtained by means of three graph operations on paths and cycles. To derive similar results in the context of other variants of domination is an open area of research.

The authors declare that they have no conflict of interests regarding the publication of this paper.

The authors are highly thankful to the anonymous referees for their kind comments and fruitful suggestions on the first draft of this paper.