Let be a non-trivial simple graph. A dominating set in graph is of vertices such that every vertex not the adjacent to at least one set. subset minimum neighborhood if and for holds. The cardinality called as number it denoted by . where intersection neighborhoods all small possible, (i.e., ). number, , In other words, smallest needed form neighborhood-dominating concept related study structure...

A rainbow colouring of a connected graph is a colouring of the edges of the graph, such that every pair of vertices is connected by at least one path in which no two edges are coloured the same. Such a colouring using minimum possible number of colours is called an optimal rainbow colouring, and the minimum number of colours required is called the rainbow connection number of the graph. A Chord...

A rainbow path in an edge coloured graph is a path in which no two edges are coloured the same. A rainbow colouring of a connected graph G is a colouring of the edges of G such that every pair of vertices in G is connected by at least one rainbow path. The minimum number of colours required to rainbow colour G is called its rainbow connection number. Between them, Chakraborty et al. [J. Comb. O...

In this paper, we introduce a new graph theoretic parameter, split edge geodetic domination number of connected as follows. A set S ⊆ V(G) is said to be dominating G if both and ( < V-S > disconnected). The minimum cardinality the called denoted by γ1gs(G). It shown that for any 3 positive integers m, f nwith 2 ≤ m n-2, there exists order n such g1 (G) = γ1gs f. For every pair l, with l γ1gs(G)...

Let R be a commutative ring with identity such that R admits at least two maximal ideals. In this article, we associate a graph with R whose vertex set is the set of all proper ideals I of R such that I is not contained in the Jacobson radical of R and distinct vertices I and J are joined by an edge if and only if I and J are not comparable under the inclusion relation. The aim of this article ...

We revisit the problem of designing a linear time algorithm for undirected graph split decomposition. Although that this problem has already been claimed to be solved in [E. Dahlhaus, FSTTCS, 1994] and [E. Dahlhaus, Journal of Algorithms 36(2):205-240, 2000], we present a new well founded theoretical background for split decomposition that allow us to clearly design and proove the rst simple li...

A graph containment problem is to decide whether one graph can be modified into some other graph by using a number of specified graph operations. We consider edge deletions, edge contractions, vertex deletions and vertex dissolutions as possible graph operations permitted. By allowing any combination of these four operations we capture the following ten problems: testing on (induced) minors, (i...

The simple max-cut problem is as follows: given a graph, find a partition of its vertex set into two disjoint sets, such that the number of edges having one endpoint in each set is as large as possible. A split graph is a graph whose vertex set admits a partition into a stable set and a clique. The simple max-cut decision problem is known to be NP-complete for split graphs. An indifference grap...

Akers et al. (Proceedings of the International Conference on Parallel Processing, 1987, pp. 393–400) proposed an interconnection topology, the star graph, as an alternative to the popular n-cube. Jwo et al. (Networks 23 (1993) 315–326) studied the alternating group graph An. Cheng et al. (Super connectivity of star graphs, alternating group graphs and split-stars, Ars Combin. 59 (2001) 107–116)...

We present a method based on simulated annealing to obtain a nested split graph that approximates a real complex graph. This is used to compute a number of graph indices using very efficient algorithms that we develop, leveraging the geometrical properties of nested split graphs. Practical results are given for six graphs from such diverse areas as social networks, communication networks, word ...