A
on a digraph
with vertex set
(
) is defined in G. Hao, X. Chen and L. Volkmann,
, Bull. Malays. Math. Sci. Soc. (2017). as a function
:
(
) → {0, 1, 2, 3} having the property that if
) = 0, then ...the vertex
must have at least two in-neighbors assigned 2 under
or one in-neighbor
with
) = 3, and if
) = 1, then the vertex
must have at least one in-neighbor
with
) ≥ 2. A set {
,
, . . .,
} of distinct double Roman dominating functions on
with the property that
for each
∈
(
) is called a double Roman dominating
(of functions) on
. The maximum number of functions in a double Roman dominating family on
is the
of
, denoted by
). We initiate the study of the double Roman domatic number, and we present different sharp bounds on
). In addition, we determine the double Roman domatic number of some classes of digraphs.
Let
be a finite and simple digraph with vertex set
(
). A signed total Roman dominating function (STRDF) on a digraph
is a function
:
(
) → {−1, 1, 2} satisfying the conditions that (i) ∑
) ≥ 1 for ...each
∈
(
), where
) consists of all vertices of
from which arcs go into
, and (ii) every vertex
for which
) = −1 has an inner neighbor
for which
) = 2. The weight of an STRDF
is
) = ∑
). The signed total Roman domination number
) of
is the minimum weight of an STRDF on
. In this paper we initiate the study of the signed total Roman domination number of digraphs, and we present different bounds on
). In addition, we determine the signed total Roman domination number of some classes of digraphs. Some of our results are extensions of known properties of the signed total Roman domination number
) of graphs
Let
be a connected graph with minimum degree
and edge-connectivity
. A graph is maximally edge-connected if
=
, and it is super-edgeconnected if every minimum edge-cut is trivial; that is, if every ...minimum edge-cut consists of edges incident with a vertex of minimum degree. The clique number
) of a graph
is the maximum cardinality of a complete subgraph of
. In this paper, we show that a connected graph
with clique number
) ≤
is maximally edge-connected or super-edge-connected if the number of edges is large enough. These are generalizations of corresponding results for triangle-free graphs by Volkmann and Hong in 2017.
If $G$ is a graph with vertex set $V(G)$, then let $Nu$ be the closed neighborhood of the vertex $u\in V(G)$. A total double Italian dominating function (TDIDF) on a graph $G$ is a function ...$f:V(G)\rightarrow\{0,1,2,3\}$ satisfying (i) $f(Nu)\ge 3$ for every vertex $u\in V(G)$ with $f(u)\in\{0,1\}$ and (ii) the subgraph induced by the vertices with a non-zero label has no isolated vertices. A TDIDF is an outer-independent total double Italian dominating function (OITDIDF) on $G$ if the set of vertices labeled $0$ induces an edgeless subgraph. The weight of an OITDIDF is the sum of its function values over all vertices, and the outer independent total double Italian domination number $\gamma_{tdI}^{oi}(G)$ is the minimum weight of an OITDIDF on $G$. In this paper, we establish various bounds on $\gamma_{tdI}^{oi}(G)$, and we determine this parameter for some special classes of graphs.
Let k ≥ 1 be an integer. A signed total Roman k-dominating function on a graph G is a function f : V (G) → {−1, 1, 2} such that Ʃ
f(u) ≥ k for every v ∈ V (G), where N(v) is the neighborhood of v, ...and every vertex u ∈ V (G) for which f(u) = −1 is adjacent to at least one vertex w for which f(w) = 2. A set {f
, f
, . . . , f
} of distinct signed total Roman k-dominating functions on G with the property that Ʃ
f
(v) ≤ k for each v ∈ V (G), is called a signed total Roman k-dominating family (of functions) on G. The maximum number of functions in a signed total Roman k-dominating family on G is the signed total Roman k-domatic number of G, denoted by d
(G). In this paper we initiate the study of signed total Roman k-domatic numbers in graphs, and we present sharp bounds for d
(G). In particular, we derive some Nordhaus-Gaddum type inequalities. In addition, we determine the signed total Roman k-domatic number of some graphs.
Total Italian domatic number of graphs Sheikholeslami, Seyed Mahmoud; Volkmann, Lutz
Computer science journal of Moldova,
07/2023, Volume:
31, Issue:
2 (92)
Journal Article
Peer reviewed
Open access
Let $G$ be a graph with vertex set $V(G)$. An \textit{Italian dominating function} (IDF) on a graph $G$ is a function $f:V(G)\longrightarrow \{0,1,2\}$ such that every vertex $v$ with $f(v)=0$ is ...adjacent to a vertex $u$ with $f(u)=2$ or to two vertices $w$ and $z$ with $f(w)=f(z)=1$. An IDF $f$ is called a \textit{total Italian dominating function} if every vertex $v$ with $f(v)\ge 1$ is adjacent to a vertex $u$ with $f(u)\ge 1$. A set $\{f_1,f_2,\ldots,f_d\}$ of distinct total Italian dominating functions on $G$ with the property that $\sum_{i=1}^df_i(v)\le 2$ for each vertex $v\in V(G)$, is called a \textit{total Italian dominating family} (of functions) on $G$. The maximum number of functions in a total Italian dominating family on $G$ is the \textit{total Italian domatic number} of $G$, denoted by $d_{tI}(G)$. In this paper, we initiate the study of the total Italian domatic number and present different sharp bounds on $d_{tI}(G)$. In addition, we determine this parameter for some classes of 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.
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