Let D be a finite and simple digraph with vertex set V (D), and let f : V (D) → {−1, 1} be a two-valued function. If k ≥ 1 is an integer and x∈N−[v] f(x) ≥ k 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 k-dominating function on D. A set {f1, f2, . . . , fd} of distinct signed k-dominating functions on D with the property that ...

Let D be a finite and simple digraph with vertex set V (D), and let f : V (D) → {−1, 1} be a two-valued function. If k ≥ 1 is an integer and ∑ x∈N−[v] f(x) ≥ k 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 k-dominating function on D. A set {f1, f2, . . . , fd} of distinct signed k-dominating functions of D with the property tha...

Let D be a finite and simple digraph with vertex set V (D), and let f : V (D)→ {−1,1} be a two-valued function. If k ≥ 1 is an integer and ∑x∈N−[v] f (x) ≥ k 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 k-dominating function on D. A set { f1, f2, . . . , fd} of distinct signed k-dominating functions of D with the property that ∑...

Roman domination is a historically inspired variety of general domination such that every vertex is labeled with labels from {0, 1, 2}. Roman domination number is the smallest of the sums of labels fulfilling condition that every vertex, labeled 0, has a neighbor, labeled 2. Using algebraic approach we give O(C) time algorithm for computing Roman domination number of special classes of polygrap...

Let G be a simple graph without isolated vertices with vertex set V (G) and edge set E(G) and let k be a positive integer. A function f : E(G) → {−1, 1} is said to be a signed star k-dominating function on G if ∑ e∈E(v) f(e) ≥ k for every vertex v of G, where E(v) = {uv ∈ E(G) | u ∈ N(v)}. A set {f1, f2, . . . , fd} of signed star k-dominating functions on G with the property that ∑d i=1 fi(e) ...

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...

Stating a two-parameter class of examples, a — positive — answer is given to the question “Are there any nontrivial hypergroups on Z?” . 1. Background and Introduction In 1989 Zeuner [14] proved that no nontrivial hypergroups (i. e. not the group) exist on the set of real numbers R. Five years later Rösler [5] showed that by weakening the axioms by considering signed hypergroups some of these s...