New Bounds on the Double Total Domination Number of Graphs

Let G be a graph of minimum degree at least two. A set $$D\subseteq V(G)$$ D ⊆ V ( G ) is said to be a double total dominating set of G if $$|N(v)\cap D|\ge 2$$ | N ( v ) ∩ D | ≥ 2 for every vertex $$v\in V(G)$$ v ∈ V ( G ) . The minimum cardinality among all double total dominating sets of G is the double total domination number of G. In this article, we continue with the study of this parameter. In particular, we provide new bounds on the double total domination number in terms of other domination parameters. Some of our results are tight bounds that improve some well-known results.

Computing locating-total domination number in some rotationally symmetric graphs

Let [Formula: see text] be a connected graph. A locating-total dominating set in a graph G is a total dominating set S of a G, for every pair of vertices [Formula: see text], such that [Formula: see text]. The minimum cardinality of a locating-total dominating set is called locating-total domination number and represented as [Formula: see text]. In this paper, locating-total domination number is determined for some cycle-related graphs. Furthermore, some well-known graphs of convex polytopes from the literature are also considered for the locating-total domination number.

Certain Properties of Domination in Product Vague Graphs With an Application in Medicine

The product vague graph (PVG) is one of the most significant issues in fuzzy graph theory, which has many applications in the medical sciences today. The PVG can manage the uncertainty, connected to the unpredictable and unspecified data of all real-world problems, in which fuzzy graphs (FGs) will not conceivably ensue into generating adequate results. The limitations of previous definitions in FGs have led us to present new definitions in PVGs. Domination is one of the highly remarkable areas in fuzzy graph theory that have many applications in medical and computer sciences. Therefore, in this study, we introduce distinctive concepts and properties related to domination in product vague graphs such as the edge dominating set, total dominating set, perfect dominating set, global dominating set, and edge independent set, with some examples. Finally, we propose an implementation of the concept of a dominating set in medicine that is related to the COVID-19 pandemic.

On k-Fair Total Domination in Graphs

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.

Hamiltonicity of 3 t EC Graphs with α = κ + 1

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 . The minimum cardinality of a total dominating set of G is the total domination number γ t G of G . The graph G is total domination edge-critical, or γ t EC , if for every edge e in the complement of G , γ t G + e < γ t G . If G is γ t EC and γ t G = k , we say that G is k t EC . In this paper, we show that every 3 t EC graph with δ G ≥ 2 and α G = κ G + 1 has a Hamilton cycle.

Bounds for complete cototal domination number of Cartesian product graphs and complement graphs

A total dominating set [Formula: see text] is said to be a complete cototal dominating set if [Formula: see text] has no isolated nodes and it is represented by [Formula: see text]. The complete cototal domination number, represented by [Formula: see text], is the minimum cardinality of a [Formula: see text] set of [Formula: see text]. In this paper, the bounds for complete cototal domination number of Cartesian product graphs and complement graphs are determined.

On Independent Transversal Dominating Sets in Graphs

A set S ⊆ V (G) is an independent transversal dominating set of a graph G if S is a dominating set of G and intersects every maximum independent set of G. An independent transversal dominating set which is a total dominating set is an independent transversal total dominating set. The minimum cardinality γit(G) (resp. γitt(G)) of an independent transversal dominating set (resp. independent transversal total dominating set) of G is the independent transversal domination number (resp. independent transversal total domination number) of G. In this paper, we show that for every positive integers a and b with 5 ≤ a ≤ b ≤ 2a − 2, there exists a connected graph G for which γit(G) = a and γitt(G) = b. We also study these two concepts in graphs which are the join, corona or composition of graphs.

Total Domination on Some Graph Operators

Let G=(V,E) be a graph; a set D⊆V is a total dominating set if every vertex v∈V has, at least, one neighbor in D. The total domination number γt(G) is the minimum cardinality among all total dominating sets. Given an arbitrary graph G, we consider some operators on this graph; S(G),R(G), and Q(G), and we give bounds or the exact value of the total domination number of these new graphs using some parameters in the original graph G.