On full friendly index sets of 1-level and 2-levels N-grids

Let G be a graph with vertex set V(G) and edge set E(G). A labeling f:V(G)→Z2 induces an edge labeling f∗:E(G)→Z2 defined by f∗(xy)=f(x)+f(y), for each edge xy∈E(G). For i∈Z2, let vf(i)=|{v∈V(G):f(v)=i}| and ef∗(i)=|{e∈E(G):f∗(e)=i}|. A labeling f of a graph G is said to be friendly if |vf(1)−vf(0)|≤1. The full friendly index set of a graph G, denoted FFI(G), is defined as {ef∗(1)−ef∗(0): the vertex labeling f is friendly}. We investigate the full friendly index sets of 1-level and 2-levels N-grids.