Majority reinforcement number

Abstract

A two-valued function [Formula: see text] defined on the vertices of a graph [Formula: see text], is a majority dominating function if the sum of its function values over at least half the closed neighborhoods is at least one. That is, for at least half the vertices [Formula: see text], [Formula: see text], where [Formula: see text] consists of [Formula: see text] and every vertex adjacent to [Formula: see text]. The majority domination number of a graph [Formula: see text], denoted [Formula: see text], is the minimum value of [Formula: see text] over all majority dominating functions [Formula: see text] of [Formula: see text]. The majority reinforcement number of [Formula: see text], denoted by [Formula: see text], is defined to be the minimum cardinality [Formula: see text] of a set [Formula: see text] of edges such that [Formula: see text]. In this paper, we initiate the study of majority reinforcement number and determine the exact values of [Formula: see text] for paths and cycles.