Bounds on the Signed 2-Independence Number in Graphs

Let $G$ be a finite and simple graph with vertex set $V (G)$, and let $f V (G) → {−1, 1}$ be a two-valued function. If $∑_{x∈N|v|} f(x) ≤ 1$ for each $v ∈ V (G)$, where $N[v]$ is the closed neighborhood of $v$, then $f$ is a signed 2-independence function on $G$. The weight of a signed 2-independence function $f$ is $w(f) = ∑_{v∈V (G)} f(v)$. The maximum of weights $w(f)$, taken over all signed 2-independence functions $f$ on $G$, is the signed 2-independence number $α_s^2(G)$ of $G$. In this work, we mainly present upper bounds on $α_s^2(G)$, as for example $α_s^2(G) ≤ n−2 [∆ (G)//2]$, and we prove the Nordhaus-Gaddum type inequality $α_s^2 (G) + α_s^2(G) ≤ n+1$, where $n$ is the order and $∆ (G)$ is the maximum degree of the graph $G$. Some of our theorems improve well-known results on the signed 2-independence number.