Let γ(G), i(G), γS(G) and iS(G) denote the domination number, the independent domination number, the strong domination number and the independent strong domination number of a graph G, respectively. A graph G is called γi-perfect (domination perfect) if γ(H) = i(H), for every induced subgraph H of G. The classes of γγS-perfect, γSiS-perfect, iiS-perfect and γiS-perfect graphs are defined analog...

Clustering can be considered as the most important unsupervised learning problem which deals with finding a structure in a collection of unlabeled data. To this end, it conducts a process of organizing objects into groups whose members are similar in some way and dissimilar to those of other groups [1]. While this process yields in an entirely unsupervised manner, additional background informat...

Many NP-hard problems can be solved efficiently when the input is restricted to graphs of bounded tree-width or cliquewidth. In particular, by the celebrated result of Courcelle, every decision problem expressible in monadic second order logic is fixed parameter tractable when parameterized by the tree-width of the input graph. On the other hand if we restrict ourselves to graphs of clique-widt...

We introduce a concept in graph coloring motivated by the popular Sudoku puzzle. Let G=(V,E) be of order n with chromatic number χ(G)=k and let S⊆V. C0 k-coloring induced subgraph G[S]. The is called an extendable if can extended to G. say that G uniquely smallest such G[S] which admits denoted sn(G). In this paper we initiate study parameter. first show parameter related list graphs. Section 2...

In this paper, we study the group and list group colorings of total graphs and present group coloring versions of the total and list total colorings conjectures.We establish the group coloring version of the total coloring conjecture for the following classes of graphs: graphs with small maximum degree, two-degenerate graphs, planner graphs with maximum degree at least 11, planner graphs withou...

In this paper we study the planar graphs that admit an acyclic 3-coloring. We show that testing acyclic 3-colorability is NP-hard, even for planar graphs of maximum degree 4, and we show that there exist infinite classes of cubic planar graphs that are not acyclically 3-colorable. Further, we show that every planar graph has a subdivision with one vertex per edge that admits an acyclic 3-colori...

We present the first distributed algorithms for computing connected dominating sets (CDS) for ad hoc networks that break the linear-time barrier. We present two algorithms which require O(∆ log n) and O(log n) running time respectively, where ∆ is the maximum node degree and n is the size of the network. This is a substantial improvement over existing implementations, all of which require Ω(n) ...

Motivated by the notion of linear coloring on simplicial complexes, recently introduced in the context of algebraic topology, and the framework through which it was studied, we introduce the colinear coloring on graphs. We provide an upper bound for the chromatic number χ(G), for any graph G, and show that G can be colinearly colored in polynomial time by proposing a simple algorithm. The colin...

This paper aims to provide a detailed survey of existing graph models and algorithms for important problems that arise in different areas of wireless telecommunication. In particular, applications of graph optimization problems such as minimum dominating set, minimum vertex coloring and maximum clique in multihop wireless networks are discussed. Different forms of graph domination have been use...