Signed Roman domination in digraphs

S. M. Sheikholeslami () and L. Volkmann ()
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S. M. Sheikholeslami: Azarbaijan Shahid Madani University
L. Volkmann: RWTH Aachen University

Journal of Combinatorial Optimization, 2015, vol. 30, issue 3, No 5, 456-467

Abstract: Abstract Let $$D$$ D be a finite and simple digraph with vertex set $$V(D)$$ V ( D ) and arc set $$A(D)$$ A ( D ) . A signed Roman dominating function (SRDF) on the digraph $$D$$ D is a function $$f:V(D)\rightarrow \{-1,1,2\}$$ f : V ( D ) → { - 1 , 1 , 2 } satisfying the conditions that (i) $$\sum _{x\in N^-[v]}f(x)\ge 1$$ ∑ x ∈ N - [ v ] f ( x ) ≥ 1 for each $$v\in V(D)$$ v ∈ V ( D ) , where $$N^-[v]$$ N - [ v ] consists of $$v$$ v and all in-neighbors of $$v$$ v , and (ii) every vertex $$u$$ u for which $$f(u)=-1$$ f ( u ) = - 1 has an in-neighbor $$v$$ v for which $$f(v)=2$$ f ( v ) = 2 . The weight of an SRDF $$f$$ f is $$w(f)=\sum _{v\in V(D)}f(v)$$ w ( f ) = ∑ v ∈ V ( D ) f ( v ) . The signed Roman domination number $$\gamma _{sR}(D)$$ γ s R ( D ) of $$D$$ D is the minimum weight of an SRDF on $$D$$ D . In this paper we initiate the study of the signed Roman domination number of digraphs, and we present different bounds on $$\gamma _{sR}(D)$$ γ s R ( D ) . In addition, we determine the signed Roman domination number of some classes of digraphs. Some of our results are extensions of well-known properties of the signed Roman domination number $$\gamma _{sR}(G)$$ γ s R ( G ) of graphs $$G$$ G .

Keywords: Digraph; Signed Roman dominating function; Signed Roman domination number; 05C20; 05C69 (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10878-013-9648-2

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