Abstract— Let G be a bipartite graph. A X-dominating set D of X of G is a strong nonsplit Xdominating set of G if every vertex in X-D is X-adjacent to all other vertices in X-D. The strong nonsplit X-domination number of a graph G, denoted by ) (G snsX γ is the minimum cardinality of a strong nonsplit X-dominating set. We find the bounds for strong nonsplit X-dominating set and give its biparti...

In this paper, we introduce the closed domination in graphs. Some interesting relationships are known between domination and closed domination and between closed domination and the independent domination. It is also shown that any triple m, k and n of positive integers with 3 ≤ m ≤ k ≤ n are realizable as the domination number, closed domination number and independent domination number, respect...

Abstract Located in the eastern section of ancient Silk Road, Hexi Corridor is a crucial area where and western civilizations met. Previous studies mainly explore human-nature interactions at particular period, there lack phased interaction long time scales. Here we present relationships patterns between humans nature region over past 10 000 years distinguish stages mechanisms interaction, whic...

Let G=(V(G),E(G)) be a graph.A set of vertices S in a graph G is called to be a Smarandachely dominating k-set, if each vertex of G is dominated by at least k vertices of S. Particularly, if k = 1, such a set is called a dominating set of G. The Smarandachely domination k -number γk(G) of G is the minimum cardinality of a Smarandachely dominating k -set of G. S is called weak domination set if ...

Let G be an IFG. Then V D  is said to bae a strong (weak) dominating set if every D V v   is strongly (weakly) dominated by some vertex in D. We denote the strong (weak) intuitionistic fuzzy dominating set by sid-set (wid-set). The minimum vertex cardinality over all the sid-set (wid-set) is called the strong (weak) dominating number of an IFG and is denoted by )] ( [ ) ( G G wid sid   In ...

Two new domination parameters for a connected graph G: the weakly convex domination number of G and the convex domination number of G are introduced. Relations between these parameters and the other domination parameters are derived. In particular, we study for which cubic graphs the convex domination number equals the connected domination number.

Topological centrality is a significant measure for characterising the relative importance of a node in a complex network. For directed networks that model dynamic processes, however, it is of more practical importance to quantify a vertex's ability to dominate (control or observe) the state of other vertices. In this paper, based on the determination of controllable and observable subspaces un...

Different topological indices will always be valuable in a variety of disciplines, including chemistry, economics, electronics, business studies, social sciences and medicine. According to the dominating set number, Wiener index for neutrosophic graphs (NG) studied considerable detail this research.
 Additionally, Average an NG has been defined, several associated theorems have introduced.

In this paper, we continue the study of the domination game in graphs introduced by Brešar, Klavžar, and Rall [SIAM J. Discrete Math. 24 (2010) 979–991]. We study the total version of the domination game and show that these two versions differ significantly. We present a key lemma, known as the Total Continuation Principle, to compare the Dominator-start total domination game and the Staller-st...