An inequality that relates the size of a bipartite graph with its order and restrained domination number

Ernst J. Joubert ()
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Ernst J. Joubert: University of Johannesburg

Journal of Combinatorial Optimization, 2016, vol. 31, issue 1, No 4, 44-51

Abstract: Abstract Let $$G=(V,E)$$ G = ( V , E ) be a graph. A set $$S\subseteq V$$ S ⊆ V is a restrained dominating set if every vertex in $$V-S$$ V - S is adjacent to a vertex in $$S$$ S and to a vertex in $$V-S$$ V - S . The restrained domination number of $$G$$ G , denoted $$\gamma _{r}(G)$$ γ r ( G ) , is the smallest cardinality of a restrained dominating set of $$G$$ G . Consider a bipartite graph $$G$$ G of order $$n\ge 4,$$ n ≥ 4 , and let $$k\in \{2,3,...,n-2\}.$$ k ∈ { 2 , 3 , . . . , n - 2 } . In this paper we will show that if $$\gamma _{r}(G)=k$$ γ r ( G ) = k , then $$m\le ((n-k)(n-k+6)+4k-8)/4$$ m ≤ ( ( n - k ) ( n - k + 6 ) + 4 k - 8 ) / 4 . We will also show that this bound is best possible.

Keywords: Bipartite graph; Domination; Restrained domination; Order; Size (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s10878-014-9709-1

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