Opuscula Math. 32, no. 1 (2012), 153-158
http://dx.doi.org/10.7494/OpMath.2012.32.1.153

 
Opuscula Mathematica

An upper bound on the total outer-independent domination number of a tree

Marcin Krzywkowski

Abstract. A total outer-independent dominating set of a graph \(G=(V(G),E(G))\) is a set \(D\) of vertices of \(G\) such that every vertex of \(G\) has a neighbor in \(D\), and the set \(V(G) \setminus D\) is independent. The total outer-independent domination number of a graph \(G\), denoted by \(\gamma_t^{oi}(G)\), is the minimum cardinality of a total outer-independent dominating set of \(G\). We prove that for every tree \(T\) of order \(n \geq 4\), with \(l\) leaves and \(s\) support vertices we have \(\gamma_t^{oi}(T) \leq (2n + s - l)/3\), and we characterize the trees attaining this upper bound.

Keywords: total outer-independent domination, total domination, tree.

Mathematics Subject Classification: 05C05, 05C69.

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Cite this article as:
Marcin Krzywkowski, An upper bound on the total outer-independent domination number of a tree, Opuscula Math. 32, no. 1 (2012), 153-158, http://dx.doi.org/10.7494/OpMath.2012.32.1.153

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