Given a graph G, we say that a subset D of the vertex set V is a dominating set if it is near all the vertices, in that every vertex outside of D is adjacent to a vertex in D. A domatic k-partition of G is a partition of V into k dominating sets. In this paper, we will consider issues of computability related to domatic partitions of computable graphs. Our investigation will center on answering...

Shephard groups are unitary reflection groups arising as the symmetries of regular complex polytopes. For a Shephard group, we identify the representation carried by the principal ideal in the coinvariant algebra generated by the image of the product of all linear forms defining reflecting hyperplanes. This representation turns out to have many equivalent guises making it analogous to the sign ...

given a graph $g$, the total dominator coloring problem seeks aproper coloring of $g$ with the additional property that everyvertex in the graph is adjacent to all vertices of a color class. weseek to minimize the number of color classes. we initiate to studythis problem on several classes of graphs, as well as findinggeneral bounds and characterizations. we also compare the totaldominator chro...

The factors influencing attitudes of Taiwan Catholics regarding sexual ethics were investigated by a survey. The number of Roman Catholics in Taiwan was estimated at 230,000 by the Roman Catholic Church in 2009. Of this total, about half are aboriginal Taiwanese, scattered through seven dioceses. The survey was sent to 405 persons in these seven dioceses, based on a .0017 ratio of the Catholic ...

A Roman dominating function (RDF) on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex v for which f(v) = 0, is adjacent to at least one vertex u for which f(u) = 2. The weight of a Roman dominating function f is the value f(V (G)) = ∑ v∈V (G) f(v). The Roman domination number of G, denoted by γR(G), is the minimum weight of an RDF on G. The Roman reinforc...

In this paper, we present new upper bounds for the global domination and Roman domination numbers and also prove that these results are asymptotically best possible. Moreover, we give upper bounds for the restrained domination and total restrained domination numbers for large classes of graphs, and show that, for almost all graphs, the restrained domination number is equal to the domination num...

A two-valued function f defined on the vertices of a graph G = (V,E), f : V → {−1, 1}, is a signed total dominating function if the sum of its function values over any open neighborhood is at least one. That is, for every v ∈ V, f(N(v)) ≥ 1, where N(v) consists of every vertex adjacent to v. The weight of a total signed dominating function is f(V ) = ∑ f(v), over all vertices v ∈ V . The total ...

We introduce joins of signed graphs and explore the chromatic number all-positive all-negative joins. prove an analogue to theorem that join two equals sum their numbers. Given graphs, is usually less than numbers, by amount depends on new concept deficiency a signed-graph coloration.

In this paper we determine the exact values of the signed domination number, signed total domination number, and minus domination number of complete multipartite graphs, which substantially generalizes some previous results obtained for special subclasses of complete multipartite graphs such as cliques and complete bipartite graphs.