In this paper, we study the Dominating Set problem in random graphs. In a random graph, each pair of vertices are joined by an edge with a probability of p, where p is a positive constant less than 1. We show that, given a random graph in n vertices, a minimum dominating set in the graph can be computed in expected 2 2 2 n) time. For the parameterized dominating set problem, we show that it can...

A function f de1ned on the vertices of a graph G = (V; E); f :V → {−1; 0; 1} is a minus dominating function if the sum of its values over any closed neighborhood is at least one. The weight of a minus dominating function is f(V ) = ∑ v∈V f(v). The minus domination number of a graph G, denoted by −(G), equals the minimum weight of a minus dominating function of G. In this paper, a sharp lower bo...

For at least partially ordered three-way tables, it is well known how to arithmetically decompose Pearson’s X2 P statistic into informative components that enable a close scrutiny of the data. Similarly well-known are smooth models for two-way tables from which score tests for homogeneity and independence can be derived. From these models, both the components of Pearson’s X2 P and information a...

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A k-distance Roman dominating function on G is a labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 has a vertex with label 2 within distance k from each other. A set {f1, f2, . . . , fd} of k-distance Roman dominating functions on G with the property that ∑d i=1 fi(v) ≤ 2 for each v ∈ V (G), is call...

An edge-coloured path in a graph is rainbow if its edges have distinct colours. The rainbow connection number of a connected graph G, denoted by rc(G), is the minimum number of colours required to colour the edges of G so that any two vertices of G are connected by a rainbow path. The function rc(G) was first introduced by Chartrand et al. [Math. Bohem., 133 (2008), pp. 85-98], and has since at...

A DD2-pair of a graph G is a pair (D,D2) of disjoint sets of vertices of G such that D is a dominating set and D2 is a 2-dominating set of G. Although there are infinitely many graphs that do not contain a DD2-pair, we show that every graph with minimum degree at least two has a DD2-pair. We provide a constructive characterization of trees that have a DD2-pair and show that K3,3 is the only con...

Given an edge-coloured graph, we say that a subgraph is rainbow if all of its edges have different colours. Let $\operatorname{ex}(n,H,$rainbow-$F)$ denote the maximal number copies $H$ properly graph on $n$ vertices can contain it has no isomorphic to $F$. We determine order magnitude $\operatorname{ex}(n,C_s,$rainbow-$C_t)$ for $s,t$ with $s\not =3$. In particular, answer question Gerbner, M\...

A function f : V (G) → {0, 1, 2} is a Roman dominating function for a graph G = (V,E) if for every vertex v with f(v) = 0, there exists a vertex w ∈ N(v) with f(w) = 2. Emperor Constantine had the requirement that an army or legion could be sent from its home to defend a neighboring location only if there was a second army which would stay and protect the home. Thus, there are two types of armi...

Rainbow vertex-connection number is the minimum k-coloring on vertex graph G and denoted by rvc(G). Besides, rainbow-vertex connection can be applied to some special graphs, such as prism path graph. Graph operation a method used create new combining two graphs. Therefore, this research uses corona product form at resulting from of (Pm,2 P3) (P3 Pm,2). The results study obtain that theorem rain...

For a given graph H and n ≥ 1, let f(n,H) denote the maximum number m for which it is possible to colour the edges of the complete graph Kn with m colours in such a way that each subgraph H in Kn has at least two edges of the same colour. Equivalently, any edge-colouring of Kn with at least rb(n,H) = f(n,H) + 1 colours contains a rainbow copy of H. The numbers f(n,H) and rb(Kn, H) are called an...