The work advances a new class of global optimization methods, called graph subdivision methods, that are based on the simultaneous adaptive subdivision of both the function’s domain of definition and the range of values.

In order to increase the paired-domination number of a graph G, minimum edges that must be subdivided (where each edge in G can no more than once) is called subdivision sdγpr(G) G. It well known sdγpr(G+e) smaller or larger for some e∉E(G). this note, we show that, if an isolated-free different from mK2, then, every e∉E(G), sdγpr(G+e)≤sdγpr(G)+2Δ(G).

Our paper has two goals: i) We propose the combinatorial approach to facilitate the calculation of the number of spanning trees for five new classes of graphs. ii) We use a new powerful operation (subdivision) to get larger graphs from a given graph. In particular, we derive the explicit formulas for the subdivision of ladder, fan, triangular snake, double triangular snake and the total graph o...

A Roman dominating function on a graph G = (V,E) is a function f : V −→ {0, 1, 2} satisfying the condition that every vertex v for which f(v) = 0 is adjacent to at least one vertex u for which f(u) = 2. The weight of a Roman dominating function is the value w(f) = ∑ v∈V f(v). The Roman domination number of a graph G, denoted by γR(G), equals the minimum weight of a Roman dominating function on ...

In 2010, Barát and Tóth verified that any r-critical graph with at most r + 4 vertices has a subdivision of Kr. Based in this result, the authors conjectured that, for every positive integer c, there exists a bound r(c) such that for any r, where r > r(c), any r-critical graph on r+ c vertices has a subdivision of Kr. In this note, we verify the validity of this conjecture for c = 5, and show c...

For every countable, connected graph A containing no one-way infinite path the following is shown: Let G be an arbitrary graph which contains for every positive integer n a system of n disjoint graphs each isomorphic to a subdivision of A. Then G also contains infinitely many disjoint subgraphs each isomorphic to a subdivision of A. In addition, corrections of errors are given that occur unfort...

In this paper, we introduced new index from the Zagreb family, named as -index is defined sum of degree five all vertices a graph. We derive some different graph operations such that Join, Cartesian product, Composition, Corona Tensor Product, Strong Disjunction, Symmetric difference, join Subdivision vertex are obtained.

This paper is an overview of what the author has learned about the critical group of a graph, including some new results. In particular we discuss the critical group of a graph in relation to that of its line graph when the original graph is regular. We begin by introducing the critical group from various aspects. We then study the subdivision graph and line graph in relation to the critical gr...

A set S of vertices in a graph G(V,E) is called a total dominating set if every vertex v ∈ V is adjacent to an element of S. The total domination number of a graph G denoted by γt(G) is the minimum cardinality of a total dominating set in G. Total domination subdivision number denoted by sdγt is the minimum number of edges that must be subdivided to increase the total domination number. Here we...

Given a graph G = (V, E), the subdivision of an edge e = uv ∈ E(G) means the substitution of the edge e by a vertex x and the new edges ux and xv. The domination subdivision number of a graph G is the minimum number of edges of G which must be subdivided (where each edge can be subdivided at most once) in order to increase the domination number. Also, the domination multisubdivision number of G...