Domination is a rapidly developing area of research in graph theory, and its various applications to ad hoc networks, distributed computing, social networks and web graphs partly explain the increased interest. This thesis focuses on domination theory, and the main aim of the study is to apply a probabilistic approach to obtain new upper bounds for various domination parameters. Chapters 2 and ...

An {em Italian dominating function} on a digraph $D$ with vertex set $V(D)$ is defined as a function$fcolon V(D)to {0, 1, 2}$ such that every vertex $vin V(D)$ with $f(v)=0$ has at least two in-neighborsassigned 1 under $f$ or one in-neighbor $w$ with $f(w)=2$. A set ${f_1,f_2,ldots,f_d}$ of distinctItalian dominating functions on $D$ with the property that $sum_{i=1}^d f_i(v)le 2$ for each $vi...

In this paper we introduce a novel, simpler form of the polytope of inner Bayesian approximations of a belief function, or “consistent probabilities”. We prove that the set of vertices of this polytope is generated by all possible permutations of elements of the domain, mirroring a similar behavior of outer consonant approximations. An intriguing connection with the behavior of maximal outer co...

As a natural variant of domination in graphs, Dankelmann et al. [Domination with exponential decay, Discrete Math. 309 (2009) 5877-5883] introduced exponential domination, where vertices are considered to have some dominating power that decreases exponentially with the distance, and the dominated vertices have to accumulate a sufficient amount of this power emanating from the dominating vertice...

For a graph [Formula: see text], double Roman dominating function (DRDF) is text] having the property that if for some vertex then has at least two neighbors assigned under or one neighbor with and text]. The weight of DRDF sum minimum on domination number denoted by bondage cardinality among all edge subsets such In this paper, we study in graphs. We determine several families graphs, present ...

A set S ⊆ V of vertices in a graph G = (V,E) is called a dominating set if every vertex in V −S is adjacent to a vertex in S. A dominating set which intersects every maximum independent set in G is called an independent transversal dominating set. The minimum cardinality of an independent transversal dominating set is called the independent transversal domination number of G and is denoted by γ...

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. In this paper, we offer a survey of selected recent results on total domination in graphs. c © 2008 Elsevier B.V. All rights reserved.

Let G = (V,E) be a graph and let S & V. The set S is a dominating set of G is every vertex of V-S is adjacent to a vertex of S. A vertex v of G is called S-perfect if \N[t~]nsi = 1 where N[v] denotes the closed neighborhood of v. The set S is defined to be a perfect neighborhood set of G if every vertex of G is S-perfect or adjacent with an S-perfect vertex. We prove that for all graphs G, O(G)...

For each vertex v in a graph G, let there be associated a subgraph Hv of G. The vertex v is said to dominate Hv as well as dominate each vertex and edge of Hv. A set S of vertices of G is called a full dominating set if every vertex of G is dominated by some vertex of S, as is every edge of G. The minimum cardinality of a full dominating set of G is its full domination number γFH(G). A full dom...