Signed 2-independence of Cartesian product of directed paths

摘要

A two-valued function f: V(D)→{-1,1) denned on the vertices of a digraph D = (V(D),A(D)) is called a signed 2-independence function (S2IF) if f(N~-[v]) ≤ 1 for every v in D. The weight of a S2IF is f(V(D)) = ∑_(ν∈V(D))f(ν). The maximum weight of a S2IF of D is the signed 2-independence number (or the lower against number) α_s~2(D) of D. Let P_m × P_n be the Cartesian product of directed paths P_m and P_n. In this paper, we determine the exact values of α_s~2(P_m × P_n) for 1 ≤ m ≤ 5 and n ≥ 1.