On k-Fair Total Domination in Graphs

Hikari, 2018

This study combines the concept of connected domination in graphs introduced by Sampathkumar and Walikar [8] in 1979 and the concept of k-fair domination in graphs introduced by Caro, Hansberg and Henning [3] in 2011. Let G = (V (G), E(G)) be a simple graph. For an integer k ≥ 1, a k-fair dominating set (kf d-set) is a dominating set S ⊆ V (G) such that |N (u) ∩ S| = k for every u ∈ V (G)\S. The k-fair domination number of G, denoted by γ kf d (G), is the minimum cardinality of a kf d-set. Let G be a connected graph. A connected k-fair dominating set (Ckf d-set) is a k-fair dominating set S ⊆ V (G) such that S , the subgraph induced by S, is connected. The smallest cardinality of a Ckf d-set in G is called the connected k-fair domination number of G, denoted by γ Ckf d (G). In this paper, we characterize the connected k-fair dominating sets in the join, corona, lexicographic and Cartesian products of graphs and determine the exact values or sharp bounds of the corresponding connected k-fair domination number.

European Journal of Pure and Applied Mathematics

In this paper, we introduce and investigate the concepts of semitotal k-fair domination and independent k-fair domination, where k is a positive integer. We also characterize the semitotal 1-fair dominating sets and independent k-fair dominating sets in the join, corona, lexicographic product, and Cartesian product of graphs and determine the exact value or sharp bounds of the corresponding semitotal 1-fair domination number and independent k-fair domination number.

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.

https://doi.org/10.30534/ijatcse/2019/44832019

A set D of vertices in a graph G (V, E) is a dominating set of G, if every vertex of V not in D is adjacent to at least one vertex in D. A dominating set D of G (V, E) is a k-fair dominating set of G, for k ≥ 1, if every vertex in V-D is adjacent to exactly k vertices in D. The k-fair domination number γ kfd (D) of G is the minimum cardinality of a k-fair dominating set. In this article, we determine the k-fair domination number of some class of graphs for k = 1.

International Journal of Mathematical Analysis, 2014

In this paper, we characterize the fair total dominating sets in the join and corona of graphs and determine the corresponding fair total domination numbers. We also characterize some fair total dominating sets in the composition of graphs and give sharp upper bounds for the corresponding fair total domination numbers.

2012

Let k be a positive integer and let G = (V,E) be a simple graph. The k-tuple domination number γ×k(G) of G is the minimum cardinality of a k-tuple dominating set S, a set that for every vertex v ∈ V, |NG[v]∩S | ≥ k. Also the total k-domination number γ×k,t(G) of G is the minimum cardi-nality of a total k-dominating set S, a set that for every vertex v ∈ V, |NG(v)∩S | ≥ k. The k-transversal number τk(H) of a hypergraph H is the minimum size of a subset S ⊆ V (H) such that |S ∩ e | ≥ k for every edge e ∈ E(H). We know that for any graph G of order n with minimum degree at least k, γ×k(G) ≤ γ×k,t(G) ≤ n. Obviously for every k-regular graph, the upper bound n is sharp. Here, we give a sufficient condition for γ×k,t(G) < n. Then we characterize complete multipartite graphs G with γ×k(G) = γ×k,t(G). We also state that the total k-domination number of a graph is the k-transversal number of its open neighborhood hypergraph, and also the domination number of a graph is the transversal num...

Pure and Applied Mathematics Quarterly, 2017

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. Keywords: k-tuple total domination number, upper k-tuple total domination number, Carte-sian and cross product graphs, hypergraph, (upper) k-transversal number. MSC(2010): 05C69.

Australas. J Comb., 2019

For a graph G, the k-total dominating graph D_{k}^{t}(G) is the graph whose vertices correspond to the total dominating sets of G that have cardinality at most k; two vertices of D_{k}^{t}(G) are adjacent if and only if the corresponding total dominating sets of G differ by either adding or deleting a single vertex. The graph D_{k}^{t}(G) is used to study the reconfiguration problem for total dominating sets: a total dominating set can be reconfigured to another by a sequence of single vertex additions and deletions, such that the intermediate sets of vertices at each step are total dominating sets, if and only if they are in the same component of D_{k}^{t}(G). Let d_{0}(G) be the smallest integer r such that D_{k}^{t}(G) is connected for all k greater than or equal to r. We investigate the realizability of graphs as total dominating graphs. For k the upper total domination number {\Gamma}_{t}(G), we show that any graph without isolated vertices is an induced subgraph of a graph G s...

European Journal of Pure and Applied Mathematics

Let G = (V (G), E(G)) be a simple graph and let α ∈ (0, 1]. A set S ⊆ V (G) isan α-partial dominating set in G if |N[S]| ≥ α |V (G)|. The smallest cardinality of an α-partialdominating set in G is called the α-partial domination number of G, denoted by ∂α(G). An α-partial dominating set S ⊆ V (G) is a total α-partial dominating set in G if every vertex in S isadjacent to some vertex in S. The total α-partial domination number of G, denoted by ∂T α(G), isthe smallest cardinality of a total α-partial dominating set in G. In this paper, we characterize thetotal partial dominating sets in the join, corona, lexicographic and Cartesian products of graphsand determine the exact values or sharp bounds of the corresponding total partial dominationnumber of these graphs.

2021

Let G = (V,E) be a simple graph. A dominating set of G is a subset D ⊆ V such that every vertex not in D is adjacent to at least one vertex in D. The cardinality of a smallest dominating set of G, denoted by γ(G), is the domination number of G. For k ≥ 1, a k-fair dominating set (kFD-set) in G, is a dominating set S such that |N(v) ∩ D| = k for every vertex v ∈ V \ D. A fair dominating set, in G is a kFD-set for some integer k ≥ 1. In this paper, after presenting preliminaries, we count the number of fair dominating sets of some specific graphs.