Average Secure Support Strong Domination in Graphs

Abstract

Let \( G = \left( {V,E} \right) \) be a simple finite and undirected graph. Let \( u \in V\left( G \right) \). Support of \( u \) denoted by \( supp\left( u \right) \) is defined as the sum of the degrees of the neighbours of \( u \). Let \( u,v \in V\left( G \right) \). \( u \) is support strongly dominated by \( v \), if \( u \) and \( v \) are adjacent and \( supp\left( v \right) \ge supp\left( u \right) \). A subset \( S \) is called support strong dominating set of \( G \), if for any \( u \in V - S \), there exists \( v \in S \) such that \( u \) and \( v \) are adjacent and \( supp\left( v \right) \ge supp\left( u \right) \) [9]. A support strong dominating set \( S \) is called a secure support strong dominating set, if for any \( u \in V - S \) there exists \( v \in S \) such that \( \left( {S - \left\{ v \right\}} \right) \cup \left\{ u \right\} \) is a support strong dominating set of \( G \). The average lower domination number \( \gamma_{av} \left( G \right) \) is defined as \( \frac{{\sum\nolimits_{v \in V\left( G \right)} {\gamma_{v} \left( G \right)} }}{{\left| {V\left( G \right)} \right|}} \), where \( \gamma_{v} \left( G \right) \) is the minimum cardinality of a minimal dominating set that contains v. The average secure support strong domination number of \( G \) denoted by \( \gamma_{sss}^{av} \left( G \right) \) is defined \( \gamma_{sss}^{av} \left( G \right) = \frac{{\sum\nolimits_{v \in V\left( G \right)} {\gamma_{sss}^{av} \left( v \right)} }}{{\left| {V\left( G \right)} \right|}} \), where \( \gamma_{sss}^{av} \left( v \right) = { \hbox{min} }\{ \left| S \right| \): S is a secure support strong dominating set of \( G \) containing \( u\} \). In this paper, average secure support strong domination number of complete \( k - ary \) tree, binomial tree are calculated. Also we obtain average secure support strong domination number of thorn graph and \( \gamma_{sss}^{av} \left( {G_{1} + G_{2} } \right) \) of \( G_{1} \) and \( G_{2} \).