The modified negative decision number in graphs (Q539336)

Summary: A mapping \(x:V\rightarrow \{- 1 , 1\} \) is called negative if \(\sum _{u \in {N\lfloor v\rfloor}}x(u) \leq 1\) . The maximum of the values of \(\sum_{v\in V}x(v) \) taken over all negative mappings \(x\), is called the modified negative decision number and is denoted by \(\beta _{v'}(G)\) . In this paper, several sharp upper bounds of this number for a general graph are presented. Exact values of these numbers for cycles, paths, cliques and bicliques are found.