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...

This paper surveys some of the work that was inspired by Wagner’s general technique to prove completeness in the levels of the boolean hierarchy over NP and some related results. In particular, we show that it is DP-complete to decide whether or not a given graph can be colored with exactly four colors, where DP is the second level of the boolean hierarchy. This result solves a question raised ...

Let D be a finite and simple digraph with the vertex set V (D), and let f : V (D) → {−1, 1} be a two-valued function. If∑ x∈N[v] f(x) ≥ 1 for each v ∈ V (D), where N[v] consists of v and all vertices of D from which arcs go into v, then f is a signed dominating function on D. The sum f(V (D)) is called the weight w(f) of f . The minimum of weights w(f), taken over all signed dominating function...

A subset, D, of the vertex set of a graph G is called a dominating set of G if each vertex of G is either in D or adjacent to some vertex in D. The maximum cardinality of a partition of the vertex set of G into dominating sets is the domatic number of G, denoted d(G). G is said to be domatically critical if the removal of any edge of G decreases the domatic number, and G is domatically full if ...

We resolve the problem posed as the main open question in [4]: letting δ(G), ∆(G) and D(G) respectively denote the minimum degree, maximum degree, and domatic number (defined below) of an undirected graph G = (V,E), we show that D(G) ≥ (1−o(1))δ(G)/ ln(∆(G)), where the “o(1)” term goes to zero as ∆(G) → ∞. A dominating set of G is any set S ⊆ V such that for all v ∈ V , either v ∈ S or some nei...

Let $D$ be a finite simple digraph with vertex set $V(D)$ and arcset $A(D)$. A twin signed total Roman dominating function (TSTRDF) on thedigraph $D$ is a function $f:V(D)rightarrow{-1,1,2}$ satisfyingthe conditions that (i) $sum_{xin N^-(v)}f(x)ge 1$ and$sum_{xin N^+(v)}f(x)ge 1$ for each $vin V(D)$, where $N^-(v)$(resp. $N^+(v)$) consists of all in-neighbors (resp.out-neighbors) of $v$, and (...

The chromatic number χ(G) of a graph G is the minimum number of colours required to colour the vertices of G in such a way that no two adjacent vertices of G receive the same colour. A partition of V into χ(G) independent sets (called colour classes) is said to be a χpartition of G. A set S ⊆ V is called a dominating set of G if every vertex in V − S is adjacent to a vertex in S. A dominating s...