Signed Roman domination in digraphs

• DOI: 10.1007/s10878-013-9648-2
• Publication Date: 2013-08-02T09:11:41Z
• Authors: S. M. Sheikholeslami; L. Volkmann
• Keywords: 0102 computer and information sciences; 01 natural sciences

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 $$\gamma _{sR}(D)$$?sR(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 $$\gamma _{sR}(D)$$?sR(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 $$\gamma _{sR}(G)$$?sR(G) of graphs $$G$$G.