Bounds on the semipaired domination number of graphs with minimum degree at least two

Teresa W. Haynes () and Michael A. Henning ()
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Teresa W. Haynes: East Tennessee State University
Michael A. Henning: University of Johannesburg

Journal of Combinatorial Optimization, 2021, vol. 41, issue 2, No 11, 486 pages

Abstract: Abstract Let G be a graph with vertex set V and no isolated vertices. A subset $$S \subseteq V$$ S ⊆ V is a semipaired dominating set of G if every vertex in $$V {\setminus } S$$ V \ S is adjacent to a vertex in S and S can be partitioned into two element subsets such that the vertices in each subset are at most distance two apart. The semipaired domination number $$\gamma _\mathrm{pr2}(G)$$ γ pr 2 ( G ) is the minimum cardinality of a semipaired dominating set of G. We show that if G is a connected graph of order n with minimum degree at least 2, then $$\gamma _\mathrm{pr2}(G) \le \frac{1}{2}(n+1)$$ γ pr 2 ( G ) ≤ 1 2 ( n + 1 ) . Further, we show that if $$n \not \equiv 3 \, (\mathrm{mod}\, 4)$$ n ≢ 3 ( mod 4 ) , then $$\gamma _\mathrm{pr2}(G) \le \frac{1}{2}n$$ γ pr 2 ( G ) ≤ 1 2 n , and we show that for every value of $$n \equiv 3 \, (\mathrm{mod}\, 4)$$ n ≡ 3 ( mod 4 ) , there exists a connected graph G of order n with minimum degree at least 2 satisfying $$\gamma _\mathrm{pr2}(G) = \frac{1}{2}(n+1)$$ γ pr 2 ( G ) = 1 2 ( n + 1 ) . As a consequence of this result, we prove that every graph G of order n with minimum degree at least 3 satisfies $$\gamma _\mathrm{pr2}(G) \le \frac{1}{2}n$$ γ pr 2 ( G ) ≤ 1 2 n .

Keywords: Domination; Paired domination; Semipaired domination; Semipaired domination number; 05C69 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10878-020-00687-w

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