A graph G is said to be 1-perfectly orientable (1-p.o. for short) if it admits an orientation such that the out-neighborhood of every vertex is a clique in G. The class of 1-p.o. graphs forms a common generalization of the classes of chordal and circular arc graphs. Even though 1-p.o. graphs can be recognized in polynomial time, no structural characterization of 1-p.o. graphs is known. In this ...

In this paper, we show that the Italian domination number of every lexicographic product graph G ○ H can be expressed in terms five different parameters . These defined under following unified approach, which encompasses definition several well-known and introduces new ones. Let N ( v ) denote open neighbourhood ∈ V , let w = 0 1 …, l a vector nonnegative integers such ≥ We say function f : → {...

Let $$w=(w_0,w_1, \dots ,w_l)$$ be a vector of nonnegative integers such that $$ w_0\ge 1$$ . G graph and N(v) the open neighbourhood $$v\in V(G)$$ We say function $$f: V(G)\longrightarrow \{0,1,\dots ,l\}$$ is w-dominating if $$f(N(v))=\sum _{u\in N(v)}f(u)\ge w_i$$ for every vertex v with $$f(v)=i$$ The weight f defined to $$\omega (f)=\sum _{v\in V(G)} f(v)$$ Given any pair adjacent vertices...

A b-coloring of the vertices of a graph is a proper coloring where each color class contains a vertex which is adjacent to each other color class. The b-chromatic number of G is the maximum integer χb(G) for which G has a b-coloring with χb(G) colors. A graph G is b-continuous if G has a b-coloring with k colors, for every integer k in the interval [χ(G), χb(G)]. It is known that not all graphs...

A homogeneous factorisation of a digraph Γ consists of a partition P = {P1, . . . , Pk} of the arc set AΓ and two vertex-transitive subgroups M 6 G 6 Aut(Γ) such that M fixes each Pi setwise while G leaves P invariant and permutes its parts transitively. Given two graphs Γ1 and Γ2 we consider several ways of taking a product of Γ1 and Γ2 to form a larger graph, namely the direct product, cartes...

Sometimes information in a network model is based on multi-agent, multi-attribute, multi-object, multi-polar information or uncertainty rather than a single bit. An m-polar fuzzy model is useful for such network models which gives more and more precision, flexibility, and comparability to the system as compared to the classical, fuzzy and bipolar fuzzy models. In this research article, we intro...

The aim of this paper is to obtain closed formulas for the perfect domination number, Roman number and lexicographic product graphs. We show that these can be obtained relatively easily case first two parameters. picture quite different when it concerns number. In case, we general bounds then give sufficient and/or necessary conditions achieved. also discuss graphs characterize where equals