A Note on Connected Interior Domination in Join and Corona of Two Graphs

IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue 2 Ver. III (Mar. - Apr. 2017), PP 66-69 www.iosrjournals.org

A Note on Connected Interior Domination in Join and Corona of Two Graphs Leomarich F. Casinillo, Emily T. Lagumbay, and Hannah Rissah F. Abad Department of Mathematics and Physics, College of Arts and Sciences, Visayas State University, Baybay City, Leyte, Philippines

___________________________________________________________________________ Abstract: Let

be a non-complete connected graph of order greater or equal to 3 with vertex set and edge set . An interior dominating set of is called a connected interior dominating set of if the subgraphs induced by is connected. The minimum cardinality of a connected interior dominating set of , denoted by , is called the connected interior domination number. A connected interior dominating set of cardinality is called a of . In this note, we revisit these concepts for some special graphs.

Further, we characterize the connected interior dominating sets in the join and corona of two graphs and give some important results.

Keywords: Non-complete connected graphs, interior dominating set, connected interior dominating set, special graphs, join, corona

I.

Introduction

Domination in graph provide numerous applications both in the position or location and protection strategies [1, 2]. This concepts was introduced by Claude Berge in 1958 [3, 5]. Also, Oystein Ore [4] introduced the terms dominating set and domination number in his book on graph theory which was published in 1962.

These concepts are helpful to find centrally located sets to cover the entire graph. The basics definitions and theorems used in this study can be found in [2, 6].

Let be a simple graph with vertex set of finite order and edge set and . The neighborhood of is the set . If then the open neighborhood of is the set The closed neighborhood of is The degree of a vertex in a graph , denoted by , is the number of edges incident with in A leaf of graph is a vertex with degree 1. A vertex that is a neighbor of a leaf is called a support.

A cut vertex is a vertex that when removed from a graph creates more components than previously in the graph .

Let and be two distinct vertices in . The distance between two vertices , of a graph is defined as the length of the shortest walk between u and v in G. If there is no walk between , then we declare The eccentricity of is the distance to a vertex farthest from , that is, The radius of a graph is the minimum eccentricity of vertex in a graph . A vertex is a central vertex if A vertex distinct from and is said to lie between and if . A vertex is an interior vertex of if for every vertex distinct from , there exists a vertex such that lies between and . A set is a dominating set of if for every , there exists such that that is, A dominating set is an interior dominating set if is dominating set of and every vertex is an interior vertex of The interior domination number of is the minimum cardinality of an interior dominating set in and is denoted by An interior dominating set of is called a connected interior dominating set of if the subgraphs induced by is connected. If then is a trivial connected interior dominating set. The minimum cardinality of a connected interior dominating set of , denoted by , is called the connected interior domination number. A connected interior dominating set of cardinality is called a of . The concepts of interior domination were introduced by Kinsley and Selvaraj [7] and connected domination by Sampathkumar and Walikar [8].

II. Preliminary Results From the definitions, the following results are immediate.

Theorem 2.1. Let be a non-complete connected graph of order

Then,

Proof. Let be a minimum interior dominating set of graph minimum connected interior dominating set of graph , that is, DOI: 10.9790/5728-1302036669

. Then, . Let be a . If the elements of are

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A Note on Connected Interior Domination in Join and Corona of Two Graphs connected interior vertices, then Otherwise connected interior dominating set, then it is easy to check that

. Thus,

. By definition of .



The upper bound in Theorem 2.1 is sharp and the lower bound is attainable.

Corollary 2.2. The difference can be made arbitrarily large.

Theorem 2.3. Let

be a

of a non-complete connected graph of order Then, i.) every cut-vertex is in and ii.) every support is in Proof. (i) Let be a cut-vertex in Suppose, for the sake of contradiction, that Since is disconnected, then it is impossible to choose . Thus, every cut-vertex is in (ii). Note that every support vertex in is also a cut-vertex in . Thus, every support is in  A leaf vertex of graph G will not be a member of interior vertex. In a tree graph, all vertices with degree greater than 1 will be a member of interior vertex.

Theorem 2.4. Let

be a non-complete connected graph of order

Then,

Proof. Let and let , where is the central vertex since vertex that dominates every vertex in Also, an interior vertex in Thus, Then, clearly  Corollary 2.5: Let , then

be a non-complete connected graph of order

with

if and only if

and a trivial connected Conversely, suppose

and

. If

The following remark is the result of connected interior domination number on some special graphs.

Remark 2.6. Let and be positive integers. Then, i.) where ii.) where iii.) where iv.) where v.) where For any complete graph vertices ,

where

, there is no connected interior dominating set, since for distinct .

III. Connected Interior Domination In The Join Of Two Graphs The join of two graphs

Lemma 3.1. If

and

is the graph

with vertex-set

is a connected interior dominating set of a graph , then

and edge-set

is a connected dominating set of

Proof. Suppose that is a connected interior dominating set of Then, is an interior dominating set of that is, is a dominating set of . Now, let . Then, there exists such that for every and the subgraph induced by is connected. Thus, is a connected dominating set of .  The converse of Lemma 3.1 is not necessarily true. We need Lemma 3.1 for the next result.

Theorem 3.2. Let and be connected graphs of order but not both complete. Then, connected interior dominating set in if and only if one of the following two statements holds: i.) is a connected dominating set of ii.) is a connected dominating set of Proof. Suppose that or . If 3.1. Thus, (i) holds. Similarly by Lemma 3.1, (ii) holds.

For the converse, suppose the statement (i) holds. Since of Let Then, there exists DOI: 10.9790/5728-1302036669

, then

is a connected dominating set of

is a

by Lemma

is a dominating set of , then is a dominating set distinct for and such that www.iosrjournals.org

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A Note on Connected Interior Domination in Join and Corona of Two Graphs . Accordingly, is a connected interior dominating set of similar manner, if (ii) holds, then is a connected interior dominating set of  Our next result characterizes all non-complete graph

of order

joined with any graph

Theorem 3.3. Let be a non-complete connected graph of order if and only if for any graph Proof. Suppose that and graph for every Conversely, suppose that It follows that for every

and let

be a positive integer. Then,

Let

Then, for any

where . Thus, and

.

Let

Then, . 

. Thus,

Corollary 3.4. Let be non-complete connected graph of order for any non-complete graph Theorem 3.5. Let be non-complete connected graph of order and , then is also dominating in

and , where and .

for every

Then,

and

Proof. Suppose that and Let . Then, is the dominating vertex in for every Since complete. It follows that is dominating set in Corollary 3.6. Let For and interior vertex in

if (i) holds. In

if and only if

, where

for every is complete, then 

, where . Then, there is no connected interior dominating set for

If

and is also

, since no

IV. Connected Interior Domination In The Corona Of Two Graphs Let and be graphs of order and , respectively. The corona of two graphs and is the graph obtained by taking one copy of and copies of , and then joining the vertex of to every vertex of the copy of For every , denote by the copy of whose vertices are attached one by one to the vertex . Denote by the subgraph of the corona corresponding to the join .

Theorem 4.1 [9] Let if and only if

be connected graph and be any graph. Then, is a dominating set of for every

is a dominating set of

We need Theorem 4.1 for the characterization of connected interior dominating sets in the corona of two graphs.

Theorem 4.2. Let be connected graph with order and be any non-complete graph. Then, is a connected interior dominating set of if and only if Proof. Suppose that

is a connected interior dominating set of . Then, by Theorem 4.1, is a dominating set of for every Since is connected and is connected with order then for every distinct from , there exists such that for every . It follows that is interior dominating vertex in . This implies that for every , for every . Then, for every .

Thus, Conversely, suppose that holds. Then, it follows that is connected since is connected with order Let for every . Then, for every , and there exists vertex distinct from such that for every .

Thus, is a connected interior dominating set of  The next corollaries and remark are direct consequence of Theorem 4.2.

Corollary 4.3. Let

be non-complete graph of order

DOI: 10.9790/5728-1302036669

. Then,

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if and only if

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A Note on Connected Interior Domination in Join and Corona of Two Graphs Proof. Let be a connected interior dominating set of . Then, is a connected interior dominating set of

by Theorem 4.2. Suppose that . Then, it follows that Thus, For the converse, let and be non-complete graph of order Suppose that .Then, and there exists such that . Thus, it is immediate that ď Ž Corollary 4.4. Let be any connected graph with Then, Remark 4.5. Let statements holds: i.) ii.)

, where

and

and

be any non-complete graph with

be any non-complete graph. Then, the following

.

The next result deals with the security of dominating set of graph Theorem 4.6. Let be any connected graph and is a dominating set of

, where

for any connected graph . If

and

Proof. Suppose that . By Theorem 4.2, and definition of corona of two graphs, is complete since is complete. It follows that minimum dominating set of . Let for every . This implies that dominating set of . Thus, is a dominating set of Corollary 4.6. Let and be any graph. Then, there exists such that

It is worth noting that

, where

and , then

. By the is the is ď Ž

and for any graph

V.

Conclusion

In this note, we obtained the bounds of connected interior domination number for any graph . We also determined the connected interior domination number of some special graphs and investigated the connected interior dominating set of join and corona of two graphs. Some of the results are characterization of join and corona of two graphs.

Acknowledgements The authors would like to acknowledge the referee for the rigorous review and giving significant comments for the improvement of this paper.

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DOI: 10.9790/5728-1302036669

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