A Roman dominating function (RDF) on a graph $G = (V, E)$ is a labeling $f : V rightarrow {0, 1, 2}$ suchthat every vertex with label $0$ has a neighbor with label $2$. The weight of $f$ is the value $f(V) = Sigma_{vin V} f(v)$The Roman domination number, $gamma_R(G)$, of $G$ is theminimum weight of an RDF on $G$.An RDF of minimum weight is called a $gamma_R$-function.A graph G is said to be $g...

Abstract A graph G is $$\alpha $$ α -excellent if every vertex of contained in some maximum independent set . In this paper, we characterize bipartite graphs, unicyclic simplicial chordal block and show that generalized Petersen -excellent.

LetG = (V,E) be a simple graph. Let u ∈ V (G). u is said to be β0 good if u is contained in a β0-set of G. u is said to beβ0-bad if there exists noβ0-set of G containing u. A graph G is said to be β0excellent if every vertex of G is β0-good. In this paper, many results on β0 excellent are obtained. And some relation between some other parameters are also investigated. Mathematics Subject Classi...

Let G be a fuzzy graph. A subset D of V is said to be Fuzzy dominating set if every vertex u ∈ V (G) there exists a vertex v ∈ V −D such that uv ∈ E(G) and μ(uv) 6 σ(u) ∧ σ(v). The minimum Cardinality of fuzzy dominating set is denoted by γf . A graph G is said to be fuzzy excellent if every vertex of G belongs to γf -sets of G. In this paper, we give a construction to imbedded non-excellent fu...

This is the first paper in a series whose goal is to give a polynomial time algorithm for the 4-coloring problem and the 4-precoloring extension problem restricted to the class of graphs with no induced six-vertex path, thus proving a conjecture of Huang. Combined with previously known results this completes the classification of the complexity of the 4-coloring problem for graphs with a connec...

This is the first paper in a series whose goal is to give a polynomial time algorithm for the $4$-coloring problem and the $4$-precoloring extension problem restricted to the class of graphs with no induced six-vertex path, thus proving a conjecture of Huang. Combined with previously known results this completes the classification of the complexity of the $4$-coloring problem for graphs with a ...

A graph $G=(V,E)$ is $\gamma$-excellent if $V$ a union of all $\gamma$-sets $G$, where $\gamma$ stands for the domination number. Let $\mathcal{I}$ be set mutually nonisomorphic graphs and $\emptyset \not= \mathcal{H} \subsetneq \mathcal{I}$. In this paper we initiate study $\mathcal{H}$-$\gamma$-excellent graphs, which define as follows. $G$ following hold: (i) every $H \in \mathcal{H}$ each $...

This is the second paper in a series of two. The goal of the series is to give a polynomial time algorithm for the 4-coloring problem and the 4-precoloring extension problem restricted to the class of graphs with no induced six-vertex path, thus proving a conjecture of Huang. Combined with previously known results this completes the classification of the complexity of the 4-coloring problem for...