Effects on fractional domination in graphs Shanthi, P.; Amutha, S.; Anbazhagan, N. ...
Journal of intelligent & fuzzy systems,
05/2023, Letnik:
44, Številka:
5
Journal Article
Recenzirano
A graph G is an undirected finite connected graph. A function f : V (G) → 0, 1 is called a fractional dominating function if, ∑u∈Nvf (u) ≥1, for all v ∈ V, where N v is the closed neighborhood of v. ...The weight of a fractional dominating function is w (f) = ∑v∈V(G)f (v). The fractional domination number γf (G) has the least weight of all the fractional dominating functions of G. In this paper, we analyze the effects on γf (G) of deleting a vertex from G. Additionally, some bounds on γf (G) are discussed, and provide the exactness of some bounds. If we remove any leaves from any tree T, then the resulting graphs are , where |l| is the number of leaves. Some of the results are proved by the eccentricity value of a vertex e (v).
Harary and Norman introduced the line graph L(G) . We introduced the legendary domination number by combining the domination concept both in graph and its line graph. In this paper, the split ...domination property is studied along with the legendary domination concept. Hence the split legendary dominating set is introduced and the corresponding split legendary domination number is defined for the line graph L(G). Also, the graph theoretical parameters are studied in terms of elements of G and its relationship with other domination parameters are presented.
We have obtained exact values for some special kinds of secure domination parameters - secure domination number, co-secure domination number, complete co-secure domination number and perfect secure ...domination number in the Sierpiński graphs and in the Hanoi graphs.
Let G be a nontrivial connected graph with vertex set V(G). A set D⊆V(G) is a double dominating set of G if |Nv∩D|≥2 for every vertex v∈V(G), where Nv represents the closed neighbourhood of v. The ...double domination number of G, denoted by γ×2(G), is the minimum cardinality among all double dominating sets of G. In this note we show that for any nontrivial tree T, n(T)−γ(T)+l(T)+s(T)+12≤γ×2(T)≤n(T)+γ(T)+l(T)2, where n(T), l(T), s(T) and γ(T) represent the order, the number of leaves, the number of support vertices and the classical domination number of T, respectively. In addition, we show that the established upper bound improves a well-known bound and as a consequence, derives two new results.
Abstract
In this paper, we introduce a new graph theoretic parameter, split edge geodetic domination number of a connected graph as follows. A set
S
⊆
V
(
G
) is said to be a split edge geodetic ...dominating set of
G
if
S
is both a split edge geodetic set and a dominating set of
G
( <
V
-
S
> is disconnected). The minimum cardinality of the split edge geodetic dominating set of
G
is called split edge geodetic domination number of
G
and is denoted by
γ
1
g
s
(
G
). It is shown that for any 3 positive integers
m
,
f
and
n
with 2 ≤
m
≤
f
≤
n
-2, there exists a connected graph
G
of order
n
such that
g
1
(
G
) =
m
and
γ
1
g
s
(
G
) =
f
. For every pair
l
,
n
of integers with 2 ≤
l
≤
n
-2, there exists a connected graph
G
of order
n
such that
γ
1
g
s
(
G
) =
l
.
Fix a positive integer n and consider the bipartite graph whose vertices are the 3-element subsets and the 2-element subsets of n={1,2,…,n}, and there is an edge between A and B if A⊂B. We prove that ...the domination number of this graph is n2−⌊(n+1)28⌋, we characterize the dominating sets of minimum size, and we observe that the minimum size dominating set can be chosen as an independent set. This is an exact version of an asymptotic result by Balogh, Katona, Linz and Tuza (2021). For the corresponding bipartite graph between the (k+1)-element subsets and the k-element subsets of n (k⩾3), we provide a new construction for small independent dominating sets. This improves on a construction by Gerbner, Kezegh, Lemons, Palmer, Pálvölgyi and Patkós (2012), who studied these independent dominating sets under the name saturating flat antichains.
Inverse Co-even Domination of Graphs Omran, Ahmed A; Shalaan, Manar M
IOP conference series. Materials Science and Engineering,
11/2020, Letnik:
928, Številka:
4
Journal Article
Recenzirano
Odprti dostop
The purpose of this paper is to introduce a new inverse domination parameter in the graphs it is called inverse co-even domination number. Some properties of the theory to this definition were only ...touched. Also, many properties and limitations on this definition are determined. Additionally, some properties of inverse co-even domination number for some certain graphs and its complement are founded, such as regular, complete, path, cycle, wheel, complete bipartite, and star.
G. Mahadevan, et, al., introduced the concept of Double Twin Domination number of a graph. DTwin (u, v) is sum of number of a u − v paths of length less than or equal to four. The total number of ...vertices that dominates every pair of vertices SDTwin(G)=∑DTwin(u,v) for u,v∈V(G) The double twin domination number of G is defined as DTD(G)=SDTwin(G)(n2). In this paper we explore this parameter for Pan, Helm, Triangular Snake, related graphs.
A set S of vertices in G is a semitotal dominating set of G if it is a dominating set of G and every vertex in S is within distance 2 of another vertex of S . The semitotal domination number , γ t 2 ...( G ), is the minimum cardinality of a semitotal dominating set of G . Clearly, γ ( G ) ≤ γ t 2 ( G ) ≤ γ t ( G ). In this paper, for any nontrivial tree T that is not a star, we investigate the ratios γ t 2 ( T )/ γ ( T ) and γ t ( T )/ γ t 2 ( T ), and provide constructive characterizations of trees achieving the upper bounds.
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