Abstract

If is a dominating set of a connected graph then the domination integrity is the minimum of the sum of two parameters, the number of elements in and the order of the maximum component of . We investigate domination integrity of splitting graph of path and cycle . This work is an effort to relate network expansion and vulnerability parameter.

1. Introduction

The stability of a communication network is of prime importance for any network designer. A shrinking network eventually loses links or nodes, its effectiveness is continuously decreasing and network becomes vulnerable. Many graph theoretic parameters have been introduced for the measurement of vulnerability. Some of them are connectivity, toughness, integrity, and rupture degree. The integrity of a graph is one of the well explored concepts which was introduced by Barefoot et al. [1].

Definition 1. The integrity of a graph is denoted by and defined by = , where is the order of a maximum component of .

Definition 2. An -set of is any (proper) subset of for which .

The integrity of the complete graph , path , cycle , star , complete bipartite graph , and power graph of cycle were discussed by Barefoot et al. [1, 2] while Goddard and Swart [3, 4] have investigated the bounds for integrity of graphs and its complement. They have also investigated the integrity of graphs in the context of some graph operations. The integrity of middle graphs is discussed by Mamut and Vumar [5] while integrity of total graphs is discussed by Dündar and Aytaç [6].

Definition 3. A set of vertices in a graph is called dominating set if every vertex is either an element of or is adjacent to an element of .

Definition 4. The domination number of a graph equals the minimum cardinality of minimal dominating set of graph .

If is any minimal dominating set and if the order of the largest component of is small, then the removal of will crash the communication network. Considering this aspect, the concept of domination integrity was introduced by Sundareswaran and Swaminathan [7].

Definition 5. The domination integrity of a connected graph is denoted as = is a dominating set}, where is the order of a maximum component of .

Sundareswaran and Swaminathan [8] have investigated domination integrity of middle graph of some graphs while Vaidya and Kothari [9] have discussed domination integrity of a graph obtained by duplication of an edge by a vertex and duplication of vertex by an edge in and . In the present work we investigate domination integrity of splitting graphs of path and cycle.

Definition 6. For a graph the splitting graph of graph is obtained by adding a new vertex corresponding to each vertex of such that where and are the neighborhood sets of and , respectively.

For standard graph theoretic notation we refer to West [10] while for terminology related to domination in Graphs we refer to Haynes et al. [11].

2. Main Results

Theorem 7. For all ,

Proof. Let be the vertices of path and be the vertices corresponding to which are added to obtain . As and , at least one vertex from each pair from and must belong to any dominating set of . Also at least one vertex from and must belong to any dominating set as and , where . Consequently
Now depending upon the number of vertices in we consider following subsets for : We claim that each is dominating set because and .
Also each is minimal because the vertex will not be dominated by any of the vertices when the vertex is removed.
Thus above defined   is  minimal  dominating  set  of  , hence from (2) we get

Observation 1. If is any dominating set of with then .

Observation 2. If is any dominating set of with then .

Theorem 8.

Proof. Let be the vertices of path and be the vertices corresponding to which are added to obtain .
Case 1 . From Theorem 7   and is a -set of . Then . This implies that since and for any dominating set of . Consequently for any dominating set of . Hence .
Case 2 . Consider the following subcases.
Subcase (i) . From Theorem 7   and is a -set of . Then . This implies that since and for any dominating set of . Consequently for any dominating set of . Hence .
Subcase (ii) . From Theorem 7   and is a -set of . Then .
Therefore If is any dominating set other than of with then . This implies that Hence from (6) and (7) .
Subcase (iii) . From Theorem 7   and is a -set of . Then .
Therefore Now if is any dominating set other than of and then . This implies that Also if is any dominating set other than and of with then .
This implies that Hence from (8), (9), and (10) .
Case 3 . From Theorem 7   and is a -set of . Then .
Therefore Now if is any dominating set other than of and then .
This implies that Also if is any dominating set other than and of with then .
This implies that Hence from (11), (12) and (13) .
Case 4 . We know that The domination number with and set with are shown in Table 1.
From Table 1 If is any dominating set of other than and with or then .
This implies that If is any dominating set other than , , and with then .
This implies that If is any dominating set other than , , , and with then .
This implies that Hence from Table 1 and the results (14) to (18) we conclude that Hence the result.

Theorem 9.

Proof. Let be the vertices of path and be the vertices corresponding to which are added to obtain . Then from Theorem 7 where for , where for , where for are -sets of . Then .
Therefore Now we claim that if for any dominating set other than then
Case 1. If is any dominating set other than and or then . Consequently .
Case 2. If is any dominating set other than and of with then . Consequently .
Case 3. If is any dominating set other than , and of with then . Consequently .
Case 4. If is any dominating set other than , , and of with then . Consequently .
Thus from (22) and (23) we have .

Theorem 10. For all

Proof. Let be the vertices of cycle and be the vertices corresponding to which are added to obtain . Now either of the vertex from and must belong to any dominating set as .
Consequently Now depending upon the different possibilities of we choose as follows. For , with (where ), for , with (where ), for , with (where ), for , with (where ).We claim that each is a minimal dominating set of since , , and and removal of , a vertex will not be dominated by any vertex hence from (25):