The Modified Negative Decision Number in Graphs

Abstract

A mappingx:V→{-1,1}is callednegativeif∑u∈N[v]x(u)≤1for everyv∈V.The maximum of the values of∑v∈Vx(v)taken over all negative mappingsx, is called themodified negative decision numberand is denoted byβD′(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.

On the inverse signed total domination number in graphs

“…Furthermore, these bounds are sharp. Also, the following sharp lower and upper bounds on STDN and ST2IN of an r-regular graph G can be found in [12] and [8,10], respectively.…”

“…The signed total domination number (STDN) of G, γ st (G), is the minimum weight of a STDF of G. If we replace "≥" and "minimum" with "≤" and "maximum", respectively, in the definition of STDN, we will have the signed total 2-independence function (ST2IF) and the signed total 2-independence number (ST2IN) of the graph, denoted by α 2 st (G). This concept was introduced in [10] and studied in [8,9] as the negative decision number.…”

On the signed 2-independence number of graphs

“…At the end of this section we exhibit a short comment about signed 2-independence number of bipartite graphs. The following upper bound on α 2 s (G) of a bipartite graph was obtained by Wang [12].…”

“…A signed 2-independence function, abbreviated S2IF, of G is defined in [14] as a function f : V → {−1, 1} such that f (N [v]) ≤ 1, for every v ∈ V . The signed 2-independence number, abbreviated S2IN, of G is α defined in [14] as a certain dual of the signed domination number of a graph [3] and has been studied by several authors including [8,10,11,12].…”

Note on the Negative Decision Number in Digraphs

“…Wang [4] presented several sharp upper bounds of the negative decision number for undirected graphs. The study of signed 2-independence number of undirected graphs was initiated by Zelink [5] and continued in [2] and elsewhere.…”