International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 135481, 9 pages
doi:10.1155/2011/135481
Research Article

The Modified Negative Decision Number in Graphs

Department of Mathematics, Ryerson University, Toronto, ON, M5B 2K3, Canada

Received 14 December 2010; Accepted 11 January 2011

Academic Editor: Dalibor Froncek

Copyright © 2011 Changping Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A mapping 𝑥 𝑉 { 1 , 1 } is called negative if 𝑢 𝑁 [ 𝑣 ] 𝑥 ( 𝑢 ) 1 for every 𝑣 𝑉 . The maximum of the values of 𝑣 𝑉 𝑥 ( 𝑣 ) taken over all negative mappings 𝑥 , is called the modified negative decision number and is denoted by 𝛽 𝐷 ( 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.