A function f : V (G) → {0, 1, 2} is a Roman dominating function if for every vertex with f(v) = 0, there exists a vertex w ∈ N(v) with f(w) = 2. We introduce two fractional Roman domination parameters, γR ◦ f and γRf , from relaxations of two equivalent integer programming formulations of Roman domination (the former using open neighborhoods and the later using closed neighborhoods in the Roman...

Domination in graphs has been an extensively researched branch of graph theory. Graph theory is one of the most flourishing branches of modern mathematics and computer applications. An introduction and an extensive overview on domination in graphs and related topics is surveyed and detailed in the two books by Haynes et al. [ 1, 2]. Recently dominating functions in domination theory have receiv...

Let G be a graph with vertex set V (G). A function f : V (G) → {−1, 1} is a signed dominating function of G if, for each vertex of G, the sum of the values of its neighbors and itself is positive. The signed domination number of a graph G, denoted γs(G), is the minimum value of ∑ v∈V (G) f(v) over all the signed dominating functions f of G. The signed reinforcement number of G, denoted Rs(G), i...

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 k > 1 is an integer and ∑ x∈N[v] f(x) > k for each v ∈ V (G), where N [v] is the closed neighborhood of v, then f is a signed k-dominating function on G. A set {f1, f2, . . . , fd} of signed kdominating functions on G with the property that ∑ d i=1 fi(x) 6 k for each x ∈ V (G), is ca...

Let D=(V(D),A(D)) be a finite, simple digraph and k positive integer. A function f:V(D)→{0,1,2,…,k+1} is called [k]-Roman dominating (for short, [k]-RDF) if f(AN−[v])≥|AN−(v)|+k for any vertex v∈V(D), where AN−(v)={u∈N−(v):f(u)≥1} AN−[v]=AN−(v)∪{v}. The weight of [k]-RDF f ω(f)=∑v∈V(D)f(v). minimum on D the domination number, denoted by γ[kR](D). For k=2 k=3, we call them double Roman number tr...

An important tool to quantify the likeness of two probability measures are f–divergences, which have seen widespread application in statistics and information theory. An example is the total variation, which plays an exceptional role among the f–divergences. It is shown that every f– divergence is bounded from below by a monotonous function of the total variation. Under appropriate regularity c...

‎Let $G=(V,E)$ be a graph‎. ‎A subset $Ssubset V$ is a hop dominating set‎‎if every vertex outside $S$ is at distance two from a vertex of‎‎$S$‎. ‎A hop dominating set $S$ which induces a connected subgraph‎ ‎is called a connected hop dominating set of $G$‎. ‎The‎‎connected hop domination number of $G$‎, ‎$ gamma_{ch}(G)$,‎‎‎ ‎is the minimum cardinality of a connected hop‎‎dominating set of $G$...