A set D ⊆ V (G) of a graph G = (V,E) is a liar’s dominating set if (1) for all v ∈ V (G) |N [v] ∩ D| ≥ 2 and (2) for every pair u, v ∈ V (G) of distinct vertices, |N [u] ∪ N [v] ∩ D| ≥ 3. In this paper, we consider the liar’s domination number of some middle graphs. Every triple dominating set is a liar’s dominating set and every liar’s dominating set must be a double dominating set. So, the li...

An {em Italian dominating function} on a digraph $D$ with vertex set $V(D)$ is defined as a function$fcolon V(D)to {0, 1, 2}$ such that every vertex $vin V(D)$ with $f(v)=0$ has at least two in-neighborsassigned 1 under $f$ or one in-neighbor $w$ with $f(w)=2$. A set ${f_1,f_2,ldots,f_d}$ of distinctItalian dominating functions on $D$ with the property that $sum_{i=1}^d f_i(v)le 2$ for each $vi...