Stability of Domination in Graphs

The stability of dominating sets in Graphs is introduced and studied,in this paper. Here D is a dominating set of Graph G. In thispaper the vertices of D and vertices of $V - D$ are called donorsand acceptors respectively. For a vertex u in D, let $\psi_{D}(u)$ denotethe number $\|N(u) \cap (V - D)\|. The donor instability or simply d-instability$d^{D}_{inst}(e)$of an edge e connecting two donor vertices v andu is$\|\psi_{D}(u)-\psi_{D}(v)\|$. The d-instability of D,$\psi_{d}(D) is the sum ofd-instabilities of all edges connecting vertices in D. For a vertex unot in D, let $\phi_{D}(u) denote the number$\|N(u)\cap D\|. The Acceptor Instabilityor simply a-instability  $a^{D}_{inst}(e)$  of an edge e connecting twoacceptor vertices u and v is $\|\phi_{D}(u)-\phi_{D}(v)\|$. The a-instability of D,$\phi_{a}(D)$ is the sum of a-instabilities of all edges connecting vertices in$V - D$. The dominating set D is d-stable if $\psi_{d}(D) = 0$ and a-stableif $\phi_{a}(D) = 0$. D is stable, if $\psi_{d}(D) = 0$ and $\psi_{a}(D) = 0$. Given anon negative integer #\alpha$, D is$\alpha-d-stable$, if$d^{D}_{inst}(e)\leq\alpha$for any edgee connecting two donor vertices and D is$\alpha-a-stable$, if$a^{D}_{inst}(e)\leq\alpha$for any edge e connecting two acceptor vertices. Here we study$\alpha$-stability number of graphs for non negative integer$\alpha$

Recognition and Combinatorial Optimization Algorithms for Bipartite Chain Graphs

In this paper we give a recognition algorithm in O(n(n+m)) time for bipartite chain graphs, and directly calculate the density of such graphs. For their stability number and domination number, we give algorithms comparable to the existing ones. We point out some applications of bipartite chain graphs in chemistry and approach the Minimum Chain Completion problem

On Topological Indices And Domination Numbers Of Graphs

Topological indices and dominating problems are popular topics in Graph Theory. There are various topological indices such as degree-based topological indices, distance-based topological indices and counting related topological indices et al. These topological indices correlate certain physicochemical properties such as boiling point, stability of chemical compounds. The concepts of domination number and independent domination number, introduced from the mid-1860s, are very fundamental in Graph Theory. In this dissertation, we provide new theoretical results on these two topics. We study k-trees and cactus graphs with the sharp upper and lower bounds of the degree-based topological indices(Multiplicative Zagreb indices). The extremal cacti with a distance-based topological index (PI index) are explored. Furthermore, we provide the extremal graphs with these corresponding topological indices. We establish and verify a proposed conjecture for the relationship between the domination number and independent domination number. The corresponding counterexamples and the graphs achieving the extremal bounds are given as well

Building Damage-Resilient Dominating Sets in Complex Networks against Random and Targeted Attacks

We study the vulnerability of dominating sets against random and targeted node removals in complex networks. While small, cost-efficient dominating sets play a significant role in controllability and observability of these networks, a fixed and intact network structure is always implicitly assumed. We find that cost-efficiency of dominating sets optimized for small size alone comes at a price of being vulnerable to damage; domination in the remaining network can be severely disrupted, even if a small fraction of dominator nodes are lost. We develop two new methods for finding flexible dominating sets, allowing either adjustable overall resilience, or dominating set size, while maximizing the dominated fraction of the remaining network after the attack. We analyze the efficiency of each method on synthetic scale-free networks, as well as real complex networks

Domination Integrity of Some Path Related Graphs

The stability of a communication network is one of the important parameters for network designers and users. A communication network can be considered to be highly vulnerable if the destruction of a few elements cause large damage and only few members are able to communicate. In a communication network several vulnerability measures like binding number, toughness, scattering number, integrity, tenacity, edge tenacity and rupture degree are used to determine the resistance of network to the disruption after the failure of certain nodes (vertices) or communication links (edges). Domination theory also provides a model to measure the vulnerability of a graph network. The domination integrity of a simple connected graph is one such measure. Here we determine the domination integrity of square graph of path as well as the graphs obtained by composition (lexicographic product) of two paths