NOTE ON THE NEGATIVE DECISION NUMBER IN DIGRAPHS

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Abstract. Let Dbe a nite digraph with the vertex set V(D) and thearc set A(D). A function f: V(D) !f 1; 1gde ned on the vertices ofa digraph Dis called a bad function if f(N (v)) 1 for every vin D.The weight of a bad function is f(V(D)) =P v2V (D) f(v). The maximumweight of a bad function of Dis the the negative decision number D (D)of D. Wang [4] studied several sharp upper bounds of this number foran undirected graph. In this paper, we study sharp upper bounds of thenegative decision number D (D) of for a digraph D. 1. IntroductionIt has been studied that an interconnection network is modelled by a graphwith vertices representing sites of the network and edges representing links be-tween sites of the network. The motivation for studying this new parameteron a directed network system may be varied from a modelling perspective.For instance, in a social network (a network of people), if we give an arc uvwhen uinuences vand assign the values -1 or 1 to the vertices of a digraph,we can model networks of people in which global decisions must be made(e.g.positive or negative responses). In certain circumstances, a positive decisioncan be made only if there are signi cantly more people voting for than thosevoting against. We assume that each individual has one vote, and each hasan initial opinion. We assign 1 to vertices (individuals) which have a positiveopinion and -1 vertices which have a negative opinion. A voter votes ’good’if there are two more vertices in its open neighborhood with positive opinionthan with negative opinion, otherwise the vote is ’bad’. We seek an assignmentof opinions that guarantee an unanimous decision; namely, for which everyvertex votes ’bad’. Such an assignment of opinions is called a uniformly nega-tive assignment. Among all uniformly negative assignments of opinions, we are