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2015•National Academy Science LettersOpen access

2-Outer-Independent Domination in Graphs

Nader Jafari Rad, Marcin Krzywkowski

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Abstract

We initiate the study of 2-outer-independent domination in graphs. A 2-outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of $$V(G) {\setminus} D$$ has at least two neighbors in D, and the set $$V(G) {\setminus} D$$ is independent. The 2-outer-independent domination number of a graph G is the minimum cardinality of a 2-outer-independent dominating set of G. We show that if a graph has minimum degree at least two, then its 2-outer-independent domination number equals the vertex cover number. Then we investigate the 2-outer-independent domination in graphs with minimum degree one.

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We initiate the study of 2-outer-independent domination in graphs. A 2-outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of $$V(G) {\setminus} D$$ has at least two neighbors in D, and the set $$V(G) {\setminus} D$$ is independent. The 2-outer-independent domination number of a graph G is the minimum cardinality of a 2-outer-independent dominating set of G. We show that if a graph has minimum degree at least two, then its 2-outer-independent domination number equals the vertex cover number. Then we investigate the 2-outer-independent domination in graphs with minimum degree one.

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Available abstract

We initiate the study of 2-outer-independent domination in graphs. A 2-outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of $$V(G) {\setminus} D$$ has at least two neighbors in D, and the set $$V(G) {\setminus} D$$ is independent. The 2-outer-independent domination number of a graph G is the minimum cardinality of a 2-outer-independent dominating set of G. We show that if a graph has minimum degree at least two, then its 2-outer-independent domination number equals the vertex cover number. Then we investigate the 2-outer-independent domination in graphs with minimum degree one.

Key concepts: Dominating set, Combinatorics, Independent set, Domination analysis, Vertex (graph theory), Maximal independent set, Mathematics, Graph

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