80 research outputs found
Signed star k-domatic number of a graph
Let be a simple graph without isolated vertices with vertex set
and edge set and let be a positive integer. A function is said to be a signed star -dominating function on if
for every vertex of , where
. A set of
signed star -dominating functions on with the property that
for each , is called a signed
star -dominating family (of functions) on . The maximum number of
functions in a signed star -dominating family on is the signed
star -domatic number of , denoted by
The signed (k,k) -domatic number of digraphs
et be a finite and simple digraph with vertex set , and
let be a two-valued function. If is an integer and for each , where consists of and all vertices of from
which arcs go into , then is a signed -dominating
function on . A set of distinct signed
-dominating functions on with the property that
for each , is called a signed
-dominating family (of functions) on . The maximum
number of functions in a signed -dominating family on
is the signed -domatic number on , denoted by
.
In this paper, we initiate the study of the signed -domatic
number of digraphs, and we present different bounds on
. Some of our results are extensions of well-known
properties of the signed domatic number of
digraphs as well as the signed -domatic number
of graphs
Outer independent total double Italian domination number
If is a graph with vertex set , then let be the closed neighborhood of the vertex . A total double Italian dominating function (TDIDF) on a graph is a function satisfying (i) for every vertex with 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 if the set of vertices labeled 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 is the minimum weight of an OITDIDF on . In this paper, we establish various bounds on , and we determine this parameter for some special classes of graphs
Total Italian domatic number of graphs
Let be a graph with vertex set . An \textit{Italian dominating function} (IDF) on a graph is a function
such that every vertex with is adjacent to a vertex with or to two vertices and with . An IDF is called a
\textit{total Italian dominating function} if every vertex with is adjacent to a vertex with .
A set of distinct total Italian dominating functions on with the property that for each vertex ,
is called a \textit{total Italian dominating family} (of functions) on . The maximum number of functions in a total Italian dominating family on is the
\textit{total Italian domatic number} of , denoted by .
In this paper, we initiate the study of the total Italian domatic number and present different sharp bounds on . In addition, we determine
this parameter for some classes of graphs
Restrained reinforcement number in graphs
A set of vertices is a restrained dominating set of a graph if every vertex in has a neighbor in and a neighbor in . The minimum cardinality of a restrained dominating set is the restrained domination number . In this paper we initiate the study of the restrained reinforcement number of a graph defined as the cardinality of a smallest set of edges for which $\gamma _{r}(G+F
RESTRAINED ROMAN REINFORCEMENT NUMBER IN GRAPHS
A restrained Roman dominating function (RRD-function) on a graph is a function from into satisfying: (i) every vertex with is adjacent to a vertex with ; (ii) the subgraph induced by the vertices assigned 0 under has no isolated vertices. The weight of an RRD-function is the sum of its function value over the whole set of vertices, and the restrained Roman domination number is the minimum weight of an RRD-function on In this paper, we begin the study of the restrained Roman reinforcement number of a graph defined as the cardinality of a smallest set of edges that we must add to the graph to decrease its restrained Roman domination number. We first show that the decision problem associated with the restrained Roman reinforcement problem is NP-hard. Then several properties as well as some sharp bounds of the restrained Roman reinforcement number are presented. In particular it is established that for every tree of order at least three
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