Bounds on inverse domination in squares of graphs

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IJRET: International Journal of Research in Engineering and Technology

eISSN: 2319-1163 | pISSN: 2321-7308

BOUNDS ON INVERSE DOMINATION IN SQUARES OF GRAPHS M. H. Muddebihal1, Srinivasa G2 1 2

Professor, Department of Mathematics, Gulbarga University, Karnataka, India, mhmuddebihal@yahoo.co.in Assistant Professor, Department of Mathematics, B. N. M. I. T , Karnataka, India, gsgraphtheory@yahoo.com

Abstract Let đ??ˇ be a minimum dominating set of a square graph đ??ş 2 . If đ?‘‰(đ??ş 2 ) − đ??ˇ contains another dominating set đ??ˇâ€˛ of đ??ş 2 , then đ??ˇâ€˛ is called an inverse dominating set with respect to đ??ˇ. The minimum cardinality of vertices in such a set is called an inverse domination number of đ??ş 2 and is denoted by đ?›ž −1 (đ??ş 2 ). In this paper, many bounds on đ?›ž −1 (đ??ş 2 ) were obtained in terms of elements of đ??ş. Also its relationship with other domination parameters was obtained.

Key words: Square graph, dominating set, inverse dominating set, Inverse domination number. Subject Classification Number: AMS-05C69, 05C70 --------------------------------------------------------------------***---------------------------------------------------------------------1. INTRODUCTION

A set đ?‘† ⊆ đ?‘‰(đ??ş) is said to be a dominating set of đ??ş, if every

In this paper, we follow the notations of [1]. We consider

vertex in (đ?‘‰ − đ?‘†) is adjacent to some vertex in đ?‘†. The

only finite undirected graphs without loops or multiple edges.

minimum cardinality of vertices in such a set is called the

In general, we use < đ?‘‹ > to denote the subgraph induced by

domination number of đ??ş and is denoted by đ?›ž đ??ş . Further, if

the set of vertices đ?‘‹ and đ?‘ đ?‘Ł and đ?‘ [đ?‘Ł] denote the open and

the subgraph < � > is independent, then � is called an

closed neighborhoods of a vertex đ?‘Ł.

independent dominating set of đ??ş. The independent omination number of đ??ş, denoted by đ?›žđ?‘– (đ??ş) is the minimum cardinality of

The minimum (maximum) degree among the vertices of đ??ş is

an independent dominating set of đ??ş.

denoted by đ?›ż đ??ş ∆ đ??ş . A vertex of degree one is called an end vertex. The term đ?›ź0 đ??ş (đ?›ź1 đ??ş ) denotes the minimum

A dominating set đ?‘† of đ??ş is said to be a connected dominating

number of vertices(edges) cover of đ??ş. Further, đ?›˝0 đ??ş (đ?›˝1 đ??ş )

set, if the subgraph < đ?‘† > is connected in đ??ş. The minimum

represents the vertex(edge) independence number of đ??ş.

cardinality of vertices in such a set is called the connected domination number of đ??ş and is denoted by đ?›žđ?‘? (đ??ş).

A vertex with degree one is called an end vertex. The distance between two vertices � and � is the length of the

A dominating set � is called total dominating set, if for every

shortest uv-path in đ??ş. The maximum distance between any

vertex đ?‘Ł ∈ đ?‘‰, there exists a vertex đ?‘˘ ∈ đ?‘†, đ?‘˘ ∉ đ?‘Ł such that

two vertices in đ??ş is called the diameter of đ??ş and is denoted

đ?‘˘ is adjacent to đ?‘Ł. The total domination number of đ??ş,

by đ?‘‘đ?‘–đ?‘Žđ?‘š(đ??ş).

denoted by đ?›žđ?‘Ą (đ??ş) is the minimum cardinality of total dominating set of đ??ş.

The square of a graph đ??ş denoted by đ??ş 2 , has the same vertices as in đ??ş and the two vertices đ?‘˘ and đ?‘Ł are joined in đ??ş 2 if and

Further, a dominating set � is called an end dominating set of

only if they are joined in đ??ş by a path of length one or two.

đ??ş, if đ?‘† contains all the end vertices in đ??ş. The minimum

The concept of squares of graphs was introduced in [2].

cardinality of vertices in such a set is called the end domination number of đ??ş and is denoted by đ?›žđ?‘’ đ??ş .

_______________________________________________________________________________________ Volume: 02 Issue: 10 | Oct-2013, Available @ http://www.ijret.org

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