Some New Results on Domination Integrity of Graphs

Abstract

The domination integrity of a connected graph G= (V(G), E(G)) is denoted as DI(G) and defined by DI(G) = min{*S*+ m(G-S) : S is a dominating set } where m(G-S) is the order of a maximum component of G-S . We discuss domination integrity in the context of some graph operations like duplication of an edge by vertex and duplication of vertex by an edge.

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Vaidya, S. and Kothari, N. (2012) Some New Results on Domination Integrity of Graphs. Open Journal of Discrete Mathematics, 2, 96-98. doi: 10.4236/ojdm.2012.23018.

1. Introduction

The vulnerability of communication network measures the resistance of network to the disruption of operation after the failure of certain station or communication links. For any communication network greater degrees of stability or less vulnerability is required. Vulnerability can be measured by certain parameters like connectivity, toughness, integrity, binding number etc. In the analysis of vulnerability of communication network to disruption, following two parameters are of great importance: 1) The size of the largest remaining group within which mutual communication can still occur; 2) The number of elements that are not functioning. In this context Barefoot et al. [1] have introduced the concept of integrity of a graph as a new measure of vulnerability of network.

Definition 1.1. The integrity of a graph G is denoted by and defined by where m(G – S) is the order of a maximum component of

Definition 1.2. An I-set of G is any (proper) subset S of for which

The connectedness of graph is not essential to define integrity. The integrity of middle graphs is discussed by Mamut and Vumar [2] while integrity of total graphs is discussed by Dundar and Aytac [3]. If D is any minimal dominating set and if the order of the largest component of G – D is small then the removal of D will crash the communication network. The decision making process as well as communication between remaining members will also be highly affected. Considering this aspect Sundareswaran and Swaminathan [4] introduced the concept of domination integrity which is defined as follows.

Definition 1.3. The domination integrity of a connected graph G is denoted as and defined by where is the order of a maximum component of.

Sundareswarn and Swaminathan [5] have investigated domination integrity of middle graph of some graphs. In the present work, we investigate the domination integrity for the graphs obtained by various graph operations. In other words we have tried to relate expansion of network with measure of vulnerability.

Definition 1.4. Duplication of a vertex by a new edge in graph G produces a new graph such that and.

Definition 1.5. Duplication of an edge by a new vertex in a graph G produces a new graph such that

Definition 1.6. For the dominating set a vertex is called isolate of if

For all other standard terminology and graph theoretical notation we refer to Hynes et al. [6].

2. Main Results

Lemma 2.1. Let G be a graph obtained by duplication of each edge by vertex of path then

Proof: Let G be a graph obtained by duplicating each edge of path by vertex There are three types of vertices in G

1)

2) for and

3)

It is obvious that must be in the dominating set as they are the most dominating vertices. From the nature of the graph G it is obvious that out of the vertices and at least one vertex must belongs to any dominating set S as is adjacent to only and Therefore if S is any dominating set then

We claim that when n is even and when is odd are minimal dominating sets.

One can observe that each and are adjacent to v2i and removal of from set S, will not be dominated by any vertex of S Hence

Theorem 2.2. Let G be a graph obtained by duplication of each edge by vertex of path then

Proof: To prove the result, we consider following three cases.

Case-1. When: Let G be a graph obtained by duplication of an edge of path by a vertex Then and as a γ-set of G and then This implies If is any dominating set with then and consequently Hence in all the cases

Case-2. When: Let G be a graph obtained by duplication of an edge v1v2 and v2v3 of path by the vertices u1 and u3 respectively. As is the only γ-set of G then and m(G – D) = 2. If S1 is any dominating set with then and. If S2 is any dominating set with then and Moreover is not possible, as the order of the largest component of is at most 3. Thus.

Case-3. When: Let G be a graph obtained by duplication of an edge of path by vertex Then with is a γ- set of and ConsequentlyIf is any dominating set withthen andIf is any dominating set with then and Hence we have.

Theorem 2.3. Let be a graph obtained by duplication of each edge by vertex of path then

(if)

Proof: Let G be a graph obtained by duplicating each edge vivi+1 of path Pn by vertex where

Then from Theorem 2.1 If n is even then is a γ-set of G otherwise is a γ-set of G Therefore which implies,

(1)

We will show that the number is minimum. For that we have to take into account the minimality of both and. The minimality of is guaranteed as S is γ-set. Now it remains to show that if S is any dominating set other than D then

If then

and

If then

which implies that

If then trivially Hence for any dominating set S

(2)

From (1) and (2) we have (if n ≥ 5).

Theorem 2.4. Let be a graph obtained by duplication of each vertex by an edge of path or cycle then

Proof: Let be a graph obtained by duplication of vertices of path or cycle by an edge Then from the construction of graph G it is obvious that from the vertices and at least one vertex must belong to any dominating set Consequently if is any dominating set then

We claim that set is a minimal dominating set. Because each is adjacent to and If is removed from set then and will not be dominated by any vertex. Thus is a minimal dominating set with minimum cardinality. Hence

Theorem 2.5. Let be a graph obtained by duplication of each vertex of path or cycle by an edge then

Proof: Let be a graph obtained by duplication of each vertex of path or cycle by an edge Then from Theorem 2.4 we have Let be a γ-set of graph G. Then Therefore,

(1)

We will show that the number is minimum. For that we have to take into account the minimality of both and The minimality of is guaranteed as S is γ-set. Now it remains to show that if S is any dominating set other than D then . If then, consequently If then trivially Hence for any dominating set S,

(2)

From (1) and (2) we have

Proposition 2.6 [6]. A dominating set is a minimal dominating set if and only if for each vertex, one of the following two conditions holds:

1) u is an isolate of.

2) There exists a vertex for which

Lemma 2.7. Let G be a graph obtained by duplication of each vertex of wheel by an edge then

Proof: Let be a graph obtained by duplication of rim vertices as well as apex vertex altogether of wheel by edges and respectively. Then each rim vertex will dominate the vertices and apex vertex For there exists a vertex such that is a singleton set. Then from Proposition 2.6 will be a minimal dominating set of If is any dominating set then we claim that Because

1) If all the elements of are only of the type then

2) If elements of are combination of and then

3) If contains any of first two types together with the apex vertex then

4) If contains and apex vertex then

Thus we have

Theorem 2.8. Let be a graph obtained by duplication of each vertex of wheel by an edge then

Proof: Let be a graph obtained by duplication of apex vertex of wheel by an edge and the rim vertices of wheel by an edge Then from Lemma 2.7 we have Let be a γ-set of graph G. Then Therefore

(1)

We will show that the number is minimum. For that we have to take into account the minimality of both and The minimality of is guaranteed as S is γ-set. Now it remains to show that if S is any dominating set other than D then

If then consequently

If then consequently

If then trivially Thus for any dominating set S,

(2)

Hence from (1) and (2)

3. Concluding Remarks

We have investigated domination integrity of three special graph families. This work relates to network expansion and measure of vulnerability. We conclude that expansion of network will provide the reason for increase of vulnerability. To investigate similar results for different graph families obtained by various graph operations is an open area of research.

4. Acknowledgements

The authors are highly thankful to the anonymous referee for valuable comments and constructive suggestions.

Conflicts of Interest

The authors declare no conflicts of interest.

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