Clique domination in a graph

Let G be an undirected graph with vertex and edge sets V (G) and E(G), respectively. A hop dominating set S ⊆ V (G) is called an outer-connected hop dominating set if S = V (G) or the subgraph ⟨V (G) \ S⟩ induced by V (G) \ S is connected. The minimum size of an outer-connected hop dominating set is the outer-connected hop domination number γfch(G). A dominating setof size γfch(G) of G is called a γfch-set. In this paper, we investigate the concept and study it for graphs resulting from some binary operations. Specifically, we characterize the outer-connected hop dominating sets in the join, corona and lexicographic products of graphs, and determine bounds of the outer-connected hop domination number of each of these graphs.

Section: Terminology and Notationmentioning

confidence: 99%

Outer-connected Hop Dominating Sets in Graphs

Saromines

Canoy²

2022

“…So far, there is a significant number of variants of hop domination that have been defined and investigated. Some studies on hop domination, its variants, and related concepts can be found in [1], [2], [5], [8], [7], [9], [13], [14], [15], [19], [20], and [21].…”

Section: Introductionmentioning

confidence: 99%

Convex Hop Domination in Graphs

Hassan

Canoy²,

Saromines³

2023

Let G be an undirected connected graph with vertex and edge sets V (G) and E(G), respectively. A set C ⊆ V (G) is called convex hop dominating if for every two vertices x, y ∈ C, the vertex set of every x-y geodesic is contained in C and for every v ∈ V (G) \ C, there exists w ∈ C such that dG(v, w) = 2. The minimum cardinality of convex hop dominating set of G, denoted by γconh(G), is called the convex hop domination number of G. In this paper, we show that every two positive integers a and b, where 2 ≤ a ≤ b, are realizable as the connected hop domination number and convex hop domination number, respectively, of a connected graph. We also characterize the convex hop dominating sets in some graphs and determine their convex hop domination numbers.

“…Domination and some variations of the concept are found in the book by Haynes et al (see [9]). Other variations of domination can be found in [2], [3], [4], [5], [11], [12], [16], and [18]. The concepts of locating, stricly locating, locating-dominating, and strictly locating-dominating, and the associated parameters are studied in [6], [8], [10], [13], [14], [15], [17], [19], [20].…”

Section: Introductionmentioning

confidence: 99%

Stable Locating-Dominating Sets in the Edge Corona and Lexicographic Product of Graphs

Malacas

Canoy²,

Chacon³

2023

A set S ⊆ V (G) of an undirected graph G is a locating-dominating set of G if for each v ∈ V (G) \ S, there exists w ∈ S such tha vw ∈ E(G) and NG(x) ∩ S ̸= NG(y) ∩ S for any two distinct vertices x and y in V (G) \ S. S is a stable locating-dominating set of G if it is a locating-dominating set of G and S \ {v} is a locating-dominating set of G for each v ∈ S. The minimum cardinality of a stable locating-dominating set of G, denoted by γSLD(G), is called the stable locating-domination number of G. In this paper, we investigate this concept and the corresponding parameter for edge corona and lexicographic product of graphs.