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On the Ratio Between 2-domination and Total Outer-Independent domination Numbers of Trees

Chinese Annals of Mathematics,Series B 2013年第5期34卷 765-776页

作者： Marcin KRZYWKOWSKI Faculty of Electronics Telecommunications and InformaticsGdan'sk University of Technology

A 2-dominating set of a graph G is a set D of vertices of G such that every vertex of V(G)\D has at least two neighbors in D.A total outer-independent dominating set of a graph 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)\D is *** 2-domination(total outer-independent domination,respectively)number of a graph G is the minimum cardinality of a 2-dominating(total outer-independent dominating,respectively)set of *** investigate the ratio between2-domination and total outer-independent domination numbers of trees.

关键词： 2-domination Total domination Total outer-independent domination,Tree

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On the 2-domination Number of Complete Grid Graphs

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Open Journal of Discrete Mathematics 2017年第1期7卷 32-50页

作者： Ramy Shaheen Suhail Mahfud Khames Almanea Department of Mathematics Faculty of Science Tishreen University Lattakia Syria

A set D of vertices of a graph G = (V, E) is called k-dominating if every vertex v ∈V-D is adjacent to some k vertices of D. The k-domination number of a graph G, γk (G), is the order of a smallest k-dominating set of G. In this paper we calculate the k-domination number (for k = 2) of the product of two paths Pm × Pn for m = 1, 2, 3, 4, 5 and arbitrary n. These results were shown an error in the paper [1].

关键词： k-Dominating Set k-domination Number 2-Dominating Set 2-domination Number Cartesian Product Graphs Paths

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On Trees with Double domination Number Equal to the 2-Outer-Independent domination Number Plus One

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Chinese Annals of Mathematics,Series B 2012年第1期33卷 113-126页

作者： Marcin KRZYWKOWSKI Faculty of Electronics Telecommunications and InformaticsGdan'sk University of TechnologyNarutowicza 11/1280-233 GdafiskPoland

A vertex of a graph is said to dominate itself and all of its neighbors. A double dominating set of a graph G is a set D of vertices of G, such that every vertex of G is dominated by at least two vertices of D. The double domination number of a graph G is the minimum cardinality of a double dominating set of G. For a graph G = （V, E）, a subset D C V（G） is a 2-dominating set if every vertex of V（G）／ D has at least two neighbors in D, while it is a 2-outer-independent dominating set of G if additionally the set V（G）／D is independent. The 2-outer-independent domination number of G is the minimum cardinality of a 2-outer-independent dominating set of G. This paper characterizes all trees with the double domination number equal to the 2-outer-independent domination number plus one.

关键词： Double domination 2-Outer-independent domination 2-domination Tree

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