ON INTEGER DOMINATION IN GRAPHS AND VIZING-LIKE PROBLEMS

A domination in graphs is part of graph theory which has many applications. Its application includes the morphological analysis, computer network communication, social network theory, CCTV installation, and many others. A set D of vertices of a simple graph G, that is a graph without loops and multiple edges, is called a dominating set if every vertex u ∈ V (G) − D is adjacent to some vertex v ∈ D. The domination number of a graph G, denoted by γ(G), is the order of a smallest dominating set of G. A dominating set D with |D| = γ(G) is called a minimum dominating set, see Haynes and Henning [5]. This research aims to find the domination number of some families of special graphs, namely Spider Web graph W b n , Helmet graph H n,m , Parachute graph P c n , and any regular graph. The results shows that the resulting domination numbers meet the lower bound of an obtained lower bound γ(G) of any graphs.

The Electronic Journal of Combinatorics, 2013

A dominating set $D$ for a graph $G$ is a subset of $V(G)$ such that any vertex not in $D$ has at least one neighbor in $D$. The domination number $\gamma(G)$ is the size of a minimum dominating set in G. Vizing's conjecture from 1968 states that for the Cartesian product of graphs $G$ and $H$, $\gamma(G)\gamma(H) \leq \gamma(G \Box H)$, and Clark and Suen (2000) proved that $\gamma(G)\gamma(H) \leq 2 \gamma(G \Box H)$. In this paper, we modify the approach of Clark and Suen to prove a variety of similar bounds related to total and paired domination, and also extend these bounds to the $n$-Cartesian product of graphs $A^1$ through $A^n$.

Discrete Mathematics, 2008

dominating function of minimum weight for a graph. In this paper, we study the variations of Ydomination such as {k}-domination, k-tuple domination, signed domination, and minus domination for some classes of graphs. We present a unified approach to these four domination problems on strongly chordal graphs. Notice that trees, block graphs, interval graphs, and directed path graphs are subclasses of strongly chordal graphs. This paper also gives complexity results for these four domination problems on doubly chordal graphs, dually chordal graphs, chordal bipartite graphs, and planar graphs. Remark 1. A graph G has a k-tuple dominating set if and only if δ(G)+1 ≥ k. A k-tuple dominating set of a graph G = (V, E) can be viewed as a k-tuple dominating function f : V → {0, 1} such that f (N G [v]) ≥ k for all vertices v ∈ V . Then, the k-tuple domination number of G is equal to the minimum weight of a k-tuple dominating function of G. Definition 2. A signed dominating function of ã 131T he 23rd Workshop on Combinatorial Mathematics and Computation Theory 132T he 23rd Workshop on Combinatorial Mathematics and Computation Theory 133T he 23rd Workshop on Combinatorial Mathematics and Computation Theory

2018

Let G = (V,E) be a simple graph. For any integer k ≥ 1, a subset of V is called a k-tuple total dominating set of G if every vertex in V has at least k neighbors in the set. The minimum cardinality of a minimal k-tuple total dominating set of G is called the k-tuple total domination number of G. In this paper, we introduce the concept of upper k-tuple total domination number of G as the maximum cardinality of a minimal k-tuple total dominating set of G, and study the problem of finding a minimal k-tuple total dominating set of maximum cardinality on several classes of graphs, as well as finding general bounds and characterizations. Also, we find some results on the upper k-tuple total domination number of the Cartesian and cross product graphs.

Journal of Combinatorial Optimization, 2009

Let γ t {k}(G) denote the total {k}-domination number of graph G, and let $G\mathbin{\square}H$ denote the Cartesian product of graphs G and H. In this paper, we show that for any graphs G and H without isolated vertices, $\gamma _{t}^{\{k\}}(G)\gamma _{t}^{\{k\}}(H)\le k(k+1)\gamma _{t}^{\{k\}}(G\mathbin{\square}H)$ . As a corollary of this result, we have $\gamma _{t}(G)\gamma _{t}(H)\le 2\gamma _{t}(G\mathbin{\square}H)$ for all graphs G and H without isolated vertices, which is given by Pak Tung Ho (Util. Math., 2008, to appear) and first appeared as a conjecture proposed by Henning and Rall (Graph. Comb. 21:63–69, 2005).

arXiv: Combinatorics, 2017

Let $G = (V, E)$ be a simple graph of order $n$. The total dominating set of $G$ is a subset $D$ of $V$ that every vertex of $V$ is adjacent to some vertices of $D$. The total domination number of $G$ is equal to minimum cardinality of total dominating set in $G$ and is denoted by $\gamma_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=\sum_{i=\gamma_t(G)}^n d_t(G,i)x^i$, where $d_t(G,i)$ is the number of total dominating sets of $G$ of size $i$. A root of $D_t(G,x)$ is called a total domination root of $G$. An irrelevant edge of $D_t(G,x)$ is an edge $e \in E$, such that $D_t(G, x) = D_t(G\setminus e, x)$. In this paper, we characterize edges possessing this property. Also we obtain some results for the number of total dominating sets of a regular graph. Finally, we study graphs with exactly two total domination roots $\{-3,0\}$, $\{-2,0\}$ and $\{-1,0\}$.

Given a graph G, a set D ⊂ V (G) is a dominating set of G if every vertex not in D is adjacent to at least one vertex of D. The domination number γ(G) is the minimum cardinality of a dominating set of G. If moreover, every vertex not in D is adjacent to exactly one vertex of D, then D is called a perfect dominating set of G. The perfect domination number γ 11 (G) is the minimum cardinality of a perfect dominating set of G. In general, for every integer k ≥ 1, a dominating set D is called a k-quasiperfect dominating set if every vertex not in D is adjacent to at most k vertices of D. The k-quasiperfect domination number γ 1k (G) is the minimum cardinality of a k-quasiperfect dominating set of G. These parameters are related in the following general way (Δ the maximum degree of G and by n the number of vertices): γ(G) = γ 1Δ (G) ≤ • • • ≤ γ 12 (G) ≤ γ 11 (G) ≤ n. In this work we study the perfect domination number, with the help of this decreasing chain of domination parameters, in the following graph families: graphs with extremal maximum degree, that is, graphs with Δ ≥ n − 3 or Δ = 3, and also in cographs, claw-free graphs and trees. We also study the behavior of these parameters under some usual product operations.

arXiv: Combinatorics, 2017

The open neighbourhood of a vertex $v$ of a graph $G$ is the set $N(v)$ consisting of all vertices adjacent to $v$ in $G$. For $D\subseteq V(G)$, we define $\overline{D}=V(G)\setminus D$. A set $D\subseteq V(G)$ is called a super dominating set of $G$ if for every vertex $u\in \overline{D}$, there exists $v\in D$ such that $N(v)\cap \overline{D}=\{u\}$. The super domination number of $G$ is the minimum cardinality among all super dominating sets in $G$. In this article, we obtain closed formulas and tight bounds for the super domination number of $G$ in terms of several invariants of $G$. Furthermore, the particular cases of corona product graphs and Cartesian product graphs are considered.

Journal of Combinatorial Optimization, 2008

In this paper we continue the investigation of total domination in Cartesian products of graphs first studied in (Henning, M.A., Rall, D.F. in Graphs Comb. 21:63-69, 2005). A set S of vertices in a graph G is a total dominating set of G if every vertex in G is adjacent to some vertex in S. The maximum cardinality of a minimal total dominating set of G is the upper total domination number of G, denoted by Γ t (G). We prove that the product of the upper total domination numbers of any graphs G and H without isolated vertices is at most twice the upper total domination number of their Cartesian product; that is, Γ t (G)Γ t (H ) ≤ 2Γ t (G H ).

European Journal of Pure and Applied Mathematics, 2021

Let G = (V (G), E(G)) be a simple non-empty graph. For an integer k ≥ 1, a k-fairtotal dominating set (kf td-set) is a total dominating set S ⊆ V (G) such that |NG(u) ∩ S| = k for every u ∈ V (G)\S. The k-fair total domination number of G, denoted by γkf td(G), is the minimum cardinality of a kf td-set. A k-fair total dominating set of cardinality γkf td(G) is called a minimum k-fair total dominating set or a γkf td-set. We investigate the notion of k-fair total domination in this paper. We also characterize the k-fair total dominating sets in the join, corona, lexicographic product and Cartesian product of graphs and determine the exact values or sharpbounds of their corresponding k-fair total domination number.