A domination graph of a digraph D, dom(D), is created using the vertex set of D and edge {u, v} ∈ E[dom(D)] whenever (u, z) ∈ A(D) or (v, z) ∈ A(D) for every other vertex z ∈ V (D). The underlying graph of a digraph D, UG(D), is the graph for which D is a biorientation. We completely characterize digraphs whose underlying graphs are identical to their domination graphs, UG(D)= dom(D). The maxim...

We generalize the concept of efficient total domination from graphs to digraphs. An efficiently total dominating set X of a digraph D is a vertex subset such that every vertex of D has exactly one predecessor in X . We study graphs that permit an orientation having such a set and give complexity results and characterizations concerning this question. Furthermore, we study the computational comp...

Let $D$ be a finite and simple digraph with vertex set $V(D)$. A weak signed Roman dominating function (WSRDF) on is $f:V(D)\rightarrow\{-1,1,2\}$ satisfying the condition that $\sum_{x\in N^-[v]}f(x)\ge 1$ for each $v\in V(D)$, where $N^-[v]$ consists of $v$ allvertices from which arcs go into $v$. The weight WSRDF $f$ $\sum_{v\in V(D)}f(v)$. domination number $\gamma_{wsR}(D)$ minimum $D$. In...

A Roman dominating function (RDF) on a digraph $D$ is a function $f: V(D)rightarrow {0,1,2}$ satisfying the condition that every vertex $v$ with $f(v)=0$ has an in-neighbor $u$ with $f(u)=2$. The weight of an RDF $f$ is the value $sum_{vin V(D)}f(v)$. The Roman domination number of a digraph $D$ is the minimum weight of an RDF on $D$. A set ${f_1,f_2,dots,f_d}$ of Roman dominating functions on ...

We introduce the price of symmetrisation, a concept that aims to compare fundamental differences (gap and quotient) between values of a given graph invariant for digraphs and the values of the same invariant of the symmetric versions of these digraphs. Basically, given some invariant our goal is to characterise digraphs that maximise price of symmetrisation. In particular, we show that for some...