Signed 2-independence in digraphs

Let D be a finite and simple digraph with vertex set V (D), and let f : V (D) -> {-1, 1} be a two-valued function If E Sigma(x is an element of N)-(vertical bar v vertical bar) f (x) <= 1 for each v is an element of V (D), where N(-)vertical bar v vertical bar consists of v and all vertices of D from which arcs go into v, then f is a signed 2-independence function on D. The sum f (V (D)) is called the weight w(f) off. The maximum of weights w(f), taken over all signed 2-independence functions f on D. is the signed 2-independence number alpha(2)(s)(D) of D. In this work, we mainly present upper bounds on alpha(2)(s)(D), as for example (1,2 (ID) n - 2 inverted right perpendicular Delta(-)/2inverted left perpendicular and alpha(2)(s)(D) <= Delta(+) + 1 - 2 inverted right perpendicular delta-/2inverted left perpendicular/Delta(+) + 1. n, where is is the order, Delta(-) and delta(-) are the maximum and the minimum indegree and Delta(+) is the maximum outdegree of the digraph D. Some of our theorems imply well-known results on the signed 2-independence number of graphs. (C) 2011 Elsevier B.V. All rights reserved.