For a graph G=(V,E) with V=V(G) and E=E(G), a Roman {3}-dominating function is a function f:V→{0,1,2,3} having the property that ∑u∈NG(v)f(u)≥3, if f(v)=0, and ∑u∈NG(v)f(u)≥2, if f(v)=1 for any ...vertex v∈G. The weight of a Roman {3}-dominating function f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a Roman {3}-dominating function on G is the Roman {3}-domination number of G, denoted by γ{R3}(G). We initiate the study of Roman {3}-domination and show its relationship to domination, Roman domination, Roman {2}-domination (Italian domination) and double Roman domination. Finally, we present an upper bound on the Roman {3}-domination number of a connected graph G in terms of the order of G and characterize the graphs attaining this bound. Finally, we show that associated decision problem for Roman {3}-domination is NP-complete, even for bipartite graphs.
Given a graph
G
=
(
V
(
G
)
,
E
(
G
)
)
, a mapping from
V
(
G
) to
{
1
,
…
,
|
V
(
G
)
|
}
(the numbers
1
,
…
,
|
V
(
G
)
|
are usually called colors) is said to be a 2-distance coloring of
G
if any ...two vertices at distance at most two from each other receive different colors. Such a mapping with the minimum number of colors is said to be an optimal 2-distance coloring of
G
. The 2-distance chromatic number
χ
2
(
G
)
of a graph
G
is the number of colors assigned by an optimal 2-distance coloring to the vertices of
G
. In this paper, we continue the study of this classic topic in graph theory. The main focus of this work is given to this parameter in some graph products, where we investigate this type of coloring in the strong, direct and rooted products. In particular, in the case of rooted products (in its most general case) we give an exact formula for the 2-distance chromatic number. This improves the results in a previous paper bounding this parameter from below and above in a special case. We next give bounds on this parameter for general graphs as well as the exact values for it in some well-known families of graphs.
For a graph
G
= (
V
,
E
), a restrained double Roman dominating function is a function
f
:
V
→ {0, 1, 2, 3} having the property that if
f
(
v
) = 0, then the vertex
v
must have at least two ...neighbors assigned 2 under
f
or one neighbor
w
with
f
(
w
) = 3, and if
f
(
v
) = 1, then the vertex
v
must have at least one neighbor
w
with
f
(
w
) ≥ 2, and at the same time, the subgraph
G
V
0 which includes vertices with zero labels has no isolated vertex. The weight of a restrained double Roman dominating function
f
is the sum
f
(
V
) = ∑
v
∈
V
f
(
v
), and the minimum weight of a restrained double Roman dominating function on
G
is the restrained double Roman domination number of
G
. We initiate the study of restrained double Roman domination with proving that the problem of computing this parameter is
NP
-hard. Then we present an upper bound on the restrained double Roman domination number of a connected graph
G
in terms of the order of
G
and characterize the graphs attaining this bound. We study the restrained double Roman domination
versus
the restrained Roman domination. Finally, we investigate the bounds for the restrained double Roman domination of trees and determine trees
T
attaining the exhibited bounds.
An outer independent double Italian dominating function on a graph $G$ is a function $f:V(G)\rightarrow\{0,1,2,3\}$ for which each vertex $x\in V(G)$ with $\color{red}{f(x)\in \{0,1\}}$ then ...$\sum_{y\in Nx}f(y)\geqslant 3$ and vertices assigned $0$ under $f$ are independent. The outer independent double Italian domination number $\gamma_{oidI}(G)$ is the minimum weight of an outer independent double Italian dominating function of graph $G$. In this work, we present some contributions to the study of outer independent double Italian domination of three graph products. We characterize the Cartesian product, lexicographic product and direct product of custom graphs in terms of this parameter. We also provide the best possible upper and lower bounds for these three products for arbitrary graphs.
For a graph G = ( V , E ) with vertex set V = V ( G ) and edge set E = E ( G ) , a Roman { 3 } -dominating function (R { 3 } -DF) is a function f : V ( G ) → { 0 , 1 , 2 , 3 } having the property ...that ∑ u ∈ N G ( v ) f ( u ) ≥ 3 , if f ( v ) = 0 , and ∑ u ∈ N G ( v ) f ( u ) ≥ 2 , if f ( v ) = 1 for any vertex v ∈ V ( G ) . The weight of a Roman { 3 } -dominating function f is the sum f ( V ) = ∑ v ∈ V ( G ) f ( v ) and the minimum weight of a Roman { 3 } -dominating function on G is the Roman { 3 } -domination number of G, denoted by γ { R 3 } ( G ) . Let G be a graph with no isolated vertices. The total Roman { 3 } -dominating function on G is an R { 3 } -DF f on G with the additional property that every vertex v ∈ V with f ( v ) ≠ 0 has a neighbor w with f ( w ) ≠ 0 . The minimum weight of a total Roman { 3 } -dominating function on G, is called the total Roman { 3 } -domination number denoted by γ t { R 3 } ( G ) . We initiate the study of total Roman { 3 } -domination and show its relationship to other domination parameters. We present an upper bound on the total Roman { 3 } -domination number of a connected graph G in terms of the order of G and characterize the graphs attaining this bound. Finally, we investigate the complexity of total Roman { 3 } -domination for bipartite graphs.
Total double Roman domination in graphs Guoliang Hao; Lutz Volkmann; Doost Ali Mojdeh
Communications in combinatorics and optimization,
06/2020, Volume:
5, Issue:
1
Journal Article
Peer reviewed
Open access
Let $G$ be a simple graph with vertex set $V$. A double Roman dominating function (DRDF) on $G$ is a function $f:V\rightarrow\{0,1,2,3\}$ satisfying that if $f(v)=0$, then the vertex $v$ must be ...adjacent to at least two vertices assigned $2$ or one vertex assigned $3$ under $f$, whereas if $f(v)=1$, then the vertex $v$ must be adjacent to at least one vertex assigned $2$ or $3$. The weight of a DRDF $f$ is the sum $\sum_{v\in V}f(v)$. A total double Roman dominating function (TDRDF) on a graph $G$ with no isolated vertex is a DRDF $f$ on $G$ with the additional property that the subgraph of $G$ induced by the set $\{v\in V:f(v)\ne0\}$ has no isolated vertices. The total double Roman domination number $\gamma_{tdR}(G)$ is the minimum weight of a TDRDF on $G$. In this paper, we give several relations between the total double Roman domination number of a graph and other domination parameters and we determine the total double Roman domination number of some classes of graphs.
Perfect double Italian domination of a graph Hao, Guoliang; Jalilolghadr, Parvin; Mojdeh, Doost Ali
AKCE international journal of graphs and combinatorics,
09/2023, Volume:
20, Issue:
3
Journal Article
Peer reviewed
Open access
AbstractFor a graph Formula: see text with Formula: see text and Formula: see text, a perfect double Italian dominating function is a function Formula: see text having the property that Formula: see ...text, for every vertex Formula: see text with Formula: see text. The weight of a perfect double Italian dominating function f is the sum Formula: see text and the minimum weight of a perfect double Italian dominating function on G is the perfect double Italian domination number Formula: see text of G. We initiate the study of perfect double Italian dominating functions. We check the Formula: see text of some standard graphs and evaluate with γdI of such graphs. The perfect double Italian dominating functions versus perfect double Roman dominating functions are perused. The NP-completeness of this parameter is verified even when it is restricted to bipartite graphs. Finally, we characterize the graphs G of order n with Formula: see text.
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