Subject Mathematics Classification: 05C38, 05C69

Let

Result 1.

Result 2.

Proof. Let

Proof. Let

(i) any non-adjacent vertex of

(ii) there exists at least 2 vertices with 2 pebbles each in

(iii) if there exists at least one vertex in

Similarly, if there exists a vertex in

Proof. Let

(i) if the adjacent vertices of

(ii) if there exists a minimum of two vertices in

(iii) if there exists a minimum of two vertices in

(iv) if there exists at least one vertex in

(v) if there exists at least one vertex in

So, consider the case where all the vertices of

Similarly, we can prove

(a)

(i)

(ii) if there exists

(b)

Proof. (a) Let

(b) Let

Proof. Consider the case where

(i)If the adjacent vertices of each

(ii)If there exists a vertex in

(iii)If there exists a minimum of 3 vertices in

(v)If there exists a minimum of 3 vertices in

Now, let us consider the case where all the vertices except one say

Similarly, we can prove

Proof. (i) Let

(i) if the adjacent vertices of

(ii) if there exists a minimum of 2 vertices in

If there exists a vertex in G with at least 8 pebbles

(iv) if there exists a minimum of 2 vertices in

Also, we can place a pebble on

(i) if the adjacent vertex of

(ii) if there exists a vertex in

Now, let us consider the case where all vertices in

Proof. Let

(i) if there exists at least two pebbles on adjacent vertices of

(ii) if there exists a minimum of two vertices in

If there exists a vertex in G with 8 pebbles

(iv) if there exists a minimum of two vertices in

(v) if there exists a vertex in

We can place a pebble on

(i) if there exists at least two pebbles on adjacent vertices of

(ii) if there exists a minimum of two vertices in

If there exists a vertex in G with 4 pebbles

(iv) if there exists a minimum of two vertices in

(v) if there exists a vertex in

Consider the case where all vertices in

Proof. Since

Proof. Let

Then

Now we use induction to show that

The result is obvious for

Therefore, we have

Then

Then

Then

Secure domination cover pebbling number is one of the areas of research in graph theory. In this paper, we have found the secure domination cover pebbling number for the join of two graphs

Given below are some interesting open problems for secure domination cover pebbling number:

Problem 1: Find the secure domination cover pebbling number for other graph operations such as sum, product, lexico product, etc.,

Problem 2: Find secure domination cover pebbling number for other families of graphs and networks.

Problem 3: Finding the complexity of secure domination cover problem.