Domatic Number Research Articles

Strong domatic number of a graph

Strong domatic number of a graph

Some Results of Total Domatic Number on Anti Fuzzy Graph

Let AG = (N, A, σ, μ) be an anti fuzzy graph. A partition DP = {D1, D2, ….., DK} of N(AG) is referred to as total domatic partition of AG if for each Di is a total dominating set of anti fuzzy graph AG and N(AG)= ⋃Di. The maximum cardinality taken over all maximum number of classes with a minimal total domatic partition of AG is called the total domatic number of AG and it is denoted by . The maximum number of classes with maximum fuzzy cardinality of a partition Di (AG) is called anti fuzzy total domatic number of anti fuzzy graph AG and it is denoted by . In this paper, we gain some preferred results and limits that referring to the full domatic number on anti fuzzy graph.

Quantum Algorithm for the Domatic Number Problem

Quantum Algorithm for the Domatic Number Problem

Graphs with minimum fractional domatic number

The domatic number of a graph is the maximum number of vertex disjoint dominating sets that partition the vertex set of the graph. In this paper we consider the fractional variant of this notion. Graphs with fractional domatic number 1 are exactly the graphs that contain an isolated vertex. Furthermore, it is known that all other graphs have fractional domatic number at least 2. In this note we characterize graphs with fractional domatic number 2. More specifically, we show that a graph without isolated vertices has fractional domatic number 2 if and only if it has a vertex of degree 1 or a connected component isomorphic to a 4-cycle. We conjecture that if the fractional domatic number is more than 2, then it is at least 7/3.

Domination-related parameters in middle graphs

The middle graph [Formula: see text] of a graph [Formula: see text] is the graph obtained by subdividing each edge of [Formula: see text] exactly once and joining all these newly introduced vertices of adjacent edges of [Formula: see text]. It is known that the decision problems for Italian domination number [Formula: see text], [Formula: see text]-rainbow domination number [Formula: see text] and Roman domination number [Formula: see text] are NP-complete. Recently, it was proved that [Formula: see text] for a graph [Formula: see text] of order [Formula: see text]. In this paper, we continue to study several domination and domatic numbers in middle graphs. We also obtain closed formulas for domination, secure domination and total domination numbers in middle graphs of rooted product graphs.

Total Italian domatic number of graphs

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.

Graphs with small Italian domatic number

Graphs with small Italian domatic number

On Domatic and Total Domatic Numbers of Cartesian Products of Graphs

On Domatic and Total Domatic Numbers of Cartesian Products of Graphs

On Domatic Number of Some Rotationally Symmetric Graphs

Domination is a well-known graph theoretic concept due to its significant real-world applications in several domains, such as design and communication network analysis, coding theory, and optimization. For a connected graph Γ = V , E , a subset U of V Γ is called a dominating set if every member present in V − U is adjacent to at least one member in U . The domatic partition is the partition of the vertices V Γ into the disjoint dominating set. The domatic number of the graph Γ is the maximum cardinality of the disjoint dominating sets. In this paper, we improved the results for the middle and central graphs of a cycle, respectively. Furthermore, we discuss the domatic number for some other cycle-related graphs and graphs of convex polytopes.

THE GLOBAL DOMINATION NUMBER IN PRODUCT FUZZY GRAPHS

In this paper we introduced the concepts of global domination number and global domatic number in product fuzzy graph and is denoted by γg(G) and dg(G), respectively we determine the global domination number γg(G) for several classes of product fuzzy graph and obtain Nordhaus-Gaddum type results for this parameter. Further, some bounds of γg(G) and dg(G) are investigated. Also the relations between γg(G)(dg(G)) and other known parameter in Product fuzzy graphs are investigated. Finally we introduce the concept of global full number and some results about this concept in product fuzzy graph are done.

On the fractional total domatic numbers of incidence graphs

<abstract><p>For a hypergraph $ H $ with vertex set $ X $ and edge set $ Y $, the incidence graph of hypergraph $ H $ is a bipartite graph $ I(H) = (X, Y, E) $, where $ xy\in E $ if and only if $ x\in X $, $ y\in Y $ and $ x\in y $. A total dominating set of graph $ G $ is a vertex subset that intersects every open neighborhood of $ G $. Let $ \mathscr{M} $ be a family of (not necessarily distinct) total dominating sets of $ G $ and $ r_{\mathscr{M}} $ be the maximum times that any vertex of $ G $ appears in $ \mathscr{M} $. The fractional domatic number $ G $ is defined as $ FTD(G) = \sup_{\mathscr{M}}\frac{|\mathscr{M}|}{r_{\mathscr{M}}} $. In 2018, Goddard and Henning showed that the incidence graph of every complete $ k $-uniform hypergraph $ H $ with order $ n $ has $ FTD(I(H)) = \frac{n}{n-k+1} $ when $ n\geq 2k\geq 4 $. We extend the result to the range $ n > k\geq 2 $. More generally, we prove that every balanced $ n $-partite complete $ k $-uniform hypergraph $ H $ has $ FTD(I(H)) = \frac{n}{n-k+1} $ when $ n\geq k $ and $ H\ncong K_n^{(n)} $, where $ FTD(I(K_n^{(n)})) = 1 $.</p></abstract>

Restrained Roman and restrained Italian domatic numbers of graphs

Let G be a graph with vertex set V ( G ) . A Roman dominating function (RDF) on a graph G is a function f : V ( G ) ⟶ { 0 , 1 , 2 } satisfying the condition that each vertex u with f ( u ) = 0 has a neighbor v with f ( v ) = 2 . An Italian dominating function (IDF) is a function f : V ( G ) ⟶ { 0 , 1 , 2 } having the property that f ( N ( u ) ) ≥ 2 for every vertex u with f ( u ) = 0 , where N ( u ) is the neighborhood of u . If f is an RDF or an IDF on G , then let V 0 = { v ∈ V ( G ) : f ( v ) = 0 } . A restrained Roman (Italian) dominating function is an RDF (IDF) f having the property that the subgraph induced by V 0 does not have an isolated vertex. A set { f 1 , f 2 , … , f d } of distinct restrained Roman (Italian) dominating functions on G with the property that ∑ i = 1 d f i ( v ) ≤ 2 for each v ∈ V ( G ) is called a restrained Roman (Italian) dominating family (of functions) on G . The maximum number of functions in a restrained Roman (Italian) dominating family on G is the restrained Roman (Italian) domatic number of G , denoted by d r R ( G ) ( d r I ( G ) ). We initiate the study of the restrained Roman (Italian) domatic numbers, and we present different sharp bounds on d r R ( G ) and d r I ( G ) . In addition, we determine these parameters for some classes of graphs.

Retraction Note to: A study on domatic number of cycle related graphs

Retraction Note to: A study on domatic number of cycle related graphs

Þ-energy of generalized Petersen graphs

ÞFor a given graph G, its Þ-energy is the sum of the absolute values of the eigenvalues of the Þ-matrix of G. In this article, we explore the Þ-energy of generalized Petersen graphs G(p,k) for various vertex partitions such as independent, domatic, total domatic and k-ply domatic partitions and partition containing a perfect matching in G(p,k). Further, we present a python program to obtain the Þ-energy of G(p,k) for the vertex partitions under consideration and examine the relation between them.

Revisiting k-tuple Dominating Sets with Emphasis on Small Values of k

For any graph $G$ of order $n$ with degree sequence $d_{1}\geq\cdots\geq d_{n}$, we define the double Slater number $s\ell_{\times2}(G)$ as the smallest integer $t$ such that $t+d_{1}+\cdots+d_{t-e}\geq2n-p$ in which $e$ and $p$ are the number of end-vertices and penultimate vertices of $G$, respectively. We show that $\gamma_{\times2}(G)\geq s\ell_{\times2}(G)$, where $\gamma_{\times2}(G)$ is the well-known double domination number of a graph $G$ with no isolated vertices. We prove that the problem of deciding whether the equality holds for a given graph is NP-complete even when restricted to $4$-partite graphs. We also prove that the problem of computing $\gamma_{\times2}(G)$ in NP-hard even for comparability graphs of diameter two. Some results concerning these two parameters are given in this paper improving and generalizing some earlier results on double domination in graphs. We give an upper bound on the $k$-tuple domatic number of graphs with characterization of all graphs attaining the bound. Finally, we characterize the family of all full graphs, leading to a solution to an open problem given in a paper by Cockayne and Hedetniemi ($1977$).

The Domatic Partition Problem in Separable Graphs

The domatic partition problem consists of partitioning a given graph into a maximum number of disjoint dominating sets. This problem is related with the domatic number problem, which consists of quantifying this maximum number of disjoint dominating sets. Both problems were proved to be NP-complete. In this paper, we present a decomposition algorithm for finding a domatic partition on separable graphs, that is, on graphs with blocks, and as a consequence, its domatic number, highly reducing the computational complexity. Computational results illustrate the benefits of the block decomposition algorithm.

Domatically full Cartesian product graphs

The domatic number [Formula: see text] of a graph [Formula: see text] is the maximum number of disjoint dominating sets in a dominating set partition of a graph [Formula: see text]. For any graph [Formula: see text], [Formula: see text] where [Formula: see text] is the minimum degree of [Formula: see text], and [Formula: see text] is domatically full if the equality holds, i.e., [Formula: see text]. In this paper, we characterize domatically full Cartesian products of a path of order 2 and a tree of order at least 3. Moreover, we show a characterization of the Cartesian product of a longer path and a tree of order at least 3. By using these results, we also show that for any two trees of order at least 3, the Cartesian product of them is domatically full.

On k-rainbow domination in middle graphs

Let G be a finite simple graph with vertex set V(G) and edge set E(G). A function f : V(G) → P({1,2,…,k}) is a k-rainbow dominating function on G if for each vertex v∈V(G) for which f(v) = ∅, it holds that ⋃u∈N(v) f(u) = {1,2,…,k}. The weight of a k-rainbow dominating function is the value ∑v∈V(G)|f(v)|. The k-rainbow domination number γrk (G) is the minimum weight of a k-rainbow dominating function on G. In this paper, we initiate the study of k-rainbow domination numbers in middle graphs. We define the concept of a middle k-rainbow dominating function, obtain some bounds related to it and determine the middle 3-rainbow domination number of some classes of graphs. We also provide upper and lower bounds for the middle 3-rainbow domination number of trees in terms of the matching number. In addition, we determine the 3-rainbow domatic number for the middle graph of paths and cycles.

Upper domatic number of regular graphs

Upper domatic number of regular graphs

Regular graphs with large Italian domatic number

For a graph $G$, an Italian dominating function is a function $f: V(G) rightarrow {0,1,2}$ such that for each vertex $v in V(G)$ either $f(v) neq 0$, or $sum_{u in N(v)} f(u) geq 2$.If a family $mathcal{F} = {f_1, f_2, dots, f_t}$ of distinct Italian dominating functions satisfy $sum^t_{i = 1} f_i(v) leq 2$ for each vertex $v$, then this is called an Italian dominating family.In [L. Volkmann, The {R}oman {${2}$}-domatic number of graphs, Discrete Appl. Math. {bf 258} (2019), 235--241], Volkmann defined the textit{Italian domatic number} of $G$, $d_{I}(G)$, as the maximum cardinality of any Italian dominating family. In this same paper, questions were raised about the Italian domatic number of regular graphs. In this paper, we show that two of the conjectures are false, and examine some exceptions to a Nordhaus-Gaddum type inequality.