Total Domatic Number Research Articles

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.

The Numerical Invariants concerning the Total Domination for Generalized Petersen Graphs

A subset S of V G is called a total dominating set of a graph G if every vertex in V G is adjacent to a vertex in S . The total domination number of a graph G denoted by γ t G is the minimum cardinality of a total dominating set in G . The maximum order of a partition of V G into total dominating sets of G is called the total domatic number of G and is denoted by d t G . Domination in graphs has applications to several fields. Domination arises in facility location problems, where the number of facilities (e.g., hospitals and fire stations) is fixed, and one attempts to minimize the distance that a person needs to travel to get to the closest facility. In this paper, the numerical invariants concerning the total domination are studied for generalized Petersen graphs.

Total Roman domatic number of a graph

A total Roman dominating function on a graph [Formula: see text] is a labeling [Formula: see text] such that every vertex with label 0 has a neighbor with label 2 and the subgraph of [Formula: see text] induced by the set of all vertices of positive weight has no isolated vertex. A set [Formula: see text] of total Roman dominating functions on [Formula: see text] with the property that [Formula: see text] for each [Formula: see text], is called a total Roman dominating family (of functions) on [Formula: see text]. The maximum number of functions in a total Roman dominating family on [Formula: see text] is the total Roman domatic number of [Formula: see text], denoted by [Formula: see text]. In this paper, we investigate the properties of total Roman domatic number in graphs. In particular, we present some sharp bounds for [Formula: see text] and we determine the total Roman domatic number of some special graphs.

Coupon-Coloring and Total Domination in Hamiltonian Planar Triangulations

We consider the so-called coupon-coloring of the vertices of a graph where every color appears in every open neighborhood, and our aim is to determine the maximal number of colors in such colorings. In other words, every color class must be a total dominating set in the graph and we study the total domatic number of the graph. We determine this parameter in every maximal outerplanar graph, and show that every Hamiltonian maximal planar graph has domatic number at least two, partially answering the conjecture of Goddard and Henning.

NP-completeness results for partitioning a graph into total dominating sets

A total domatic k-partition of a graph is a partition of its vertex set into k subsets such that each intersects the open neighborhood of each vertex. The maximum k for which a total domatic k-partition exists is known as the total domatic number of a graph G, denoted by dt(G). We extend considerably the known hardness results by showing it is ▪-complete to decide whether dt(G)≥3 where G is a bipartite planar graph of bounded maximum degree. Similarly, for every k≥3, it is ▪-complete to decide whether dt(G)≥k, where G is split or k-regular. In particular, these results complement recent combinatorial results regarding dt(G) on some of these graph classes by showing that the known results are, in a sense, best possible. Finally, for general n-vertex graphs, we show the problem is solvable in 2nnO(1) time, and derive even faster algorithms for special graph classes.

Cubic graphs with total domatic number at least two

Let $G$ be a graph. A total dominating set of $G$ is a set $S$ of vertices of $G$ such that every vertex is adjacent to at least one vertex in $S$. The total domatic number of a graph is the maximum number of total dominating sets which partition the vertex set of $G$. In this paper we would like to characterize the cubic graphs with total domatic number at least two.

Signed Total Domatic Number of Directed Circulant Graphs

Signed Total Domatic Number of Directed Circulant Graphs

Perfectly relating the domination, total domination, and paired domination numbers of a graph

The domination number γ(G), the total domination number γt(G), the paired domination number γp(G), the domatic number d(G), and the total domatic number dt(G) of a graph G without isolated vertices are related by trivial inequalities γ(G)≤γt(G)≤γp(G)≤2γ(G) and dt(G)≤d(G). Very little is known about the graphs that satisfy one of these inequalities with equality. We study classes of graphs defined by requiring equality in one of the above inequalities for all induced subgraphs that have no isolated vertices and whose domination number is not too small. Our results are characterizations of several such classes in terms of their minimal forbidden induced subgraphs. Furthermore, we prove some hardness results, which suggest that the extremal graphs for some of the above inequalities do not have a simple structure.

On the total domatic number of regular graphs

A set S of vertices of a graph G = (V, E) without isolated vertex is a total dominating set if every vertex of V (G) is adjacent to some vertex in S. The total domatic number of a graph G is the maximum number of total dominating sets into which the vertex set of G can be partitioned. We show that the total domatic number of a random r-regular graph is almost surely at most r − 1, and that for 3-regular random graphs, the total domatic number is almost surely equal to 2. We also give a lower bound on the total domatic number of a graph in terms of order, minimum degree and maximum degree. As a corollary, we obtain the result that the total domatic number of an r-regular graph is at least r/(3 ln(r)).

SIGNED TOTAL {K}-DOMINATION AND {K}-DOMATIC NUMBERS OF GRAPHS

Let k be a positive integer, and let G be a simple graph with vertex set V(G). A function f : V(G) → {±1, ±2, …, ±k} is called a signed total {k}-dominating function if ∑u∈N(v) f(u) ≥ k for each vertex v ∈ V(G). A set {f1, f2, …, fd} of signed total {k}-dominating functions on G with the property that [Formula: see text] for each v∈V(G), is called a signed total {k}-dominating family (of functions) on G. The maximum number of functions in a signed total {k}-dominating family on G is the signed total {k}-domatic number of G, denoted by [Formula: see text]. Note that [Formula: see text] is the classical signed total domatic number dS(G). In this paper, we initiate the study of signed total k-domatic numbers in graphs, and we present some sharp upper bounds for [Formula: see text]. In addition, we determine [Formula: see text] for several classes of graphs. Some of our results are extensions of known properties of the signed total domatic number.

Signed total (k, k)-domatic number of digraphs

Let D be a finite and simple digraph with vertex set V(D), and let f: V(D) → {−1, 1} be a two-valued function. If k ≥ 1 is an integer and \({\sum_{x \in N^-(v)}f(x) \ge k}\) for each \({v \in V(G)}\), where N −(v) consists of all vertices of D from which arcs go into v, then f is a signed total k-dominating function on D. A set {f 1, f 2, . . . , f d } of signed total k-dominating functions on D with the property that \({\sum_{i=1}^df_i(x)\le k}\) for each \({x \in V(D)}\) , is called a signed total (k, k)-dominating family (of functions) on D. The maximum number of functions in a signed total (k, k)-dominating family on D is the signed total (k, k)-domatic number on D, denoted by \({d_{st}^{k}(D)}\) . In this paper we initiate the study of the signed total (k, k)-domatic number of digraphs, and we present different bounds on \({d_{st}^{k}(D)}\) . Some of our results are extensions of known properties of the signed total domatic number \({d_{st}(D)=d_{st}^{1}(D)}\) of digraphs D as well as the signed total domatic number d st (G) of graphs G, given by Henning (Ars Combin. 79:277–288, 2006).

Upper bounds on the signed total domatic number of graphs

Let G be a graph with the vertex set V ( G ) , and let f : V ( G ) → { − 1 , 1 } be a two-valued function. If G has no isolated vertices and ∑ x ∈ N ( v ) f ( x ) ≥ 1 for each v ∈ V ( G ) , where N ( v ) is the neighborhood of v , then f is a signed total dominating function on G . A set { f 1 , f 2 , … , f d } of signed total dominating functions on G with the property that ∑ i = 1 d f i ( x ) ≤ 1 for each x ∈ V ( G ) is called a signed total dominating family (of functions) on G . The maximum number of functions in a signed total dominating family on G is the signed total domatic number of G , denoted by d t S ( G ) . In this article we mainly present upper bounds on d t S ( G ) , in particular for regular graphs. As an application of these bounds, we show that d t S ( G ) ≤ n − 3 for any graph G of order n ≥ 4 without isolated vertices. Furthermore, we prove the Nordhaus–Gaddum inequality d t S ( G ) + d t S ( G ¯ ) ≤ n − 3 for graphs G and G ¯ of order n ≥ 7 without isolated vertices, where G ¯ is the complement of G .

Neighbourhood total domination in graphs

Let G = (V;E) be a graph without isolated vertices. A dominating set S of G is called a neighbourhood total dominating set (ntd-set) if the induced subgraphhN(S)i has no isolated vertices. The minimum cardinality of a ntd-set of G is called the neighbourhood total domination number of G and is denoted by nt(G). The maximum order of a partition of V into ntd-sets is called the neighbourhood total domatic number of G and is denoted by dnt(G): In this paper we initiate a study of these parameters.

The total {k}-domatic number of a graph

For a positive integer k, a total {k}-dominating function of a graph G is a function f from the vertex set V(G) to the set {0,1,2,?,k} such that for any vertex v?V(G), the condition ? u?N(v) f(u)?k is fulfilled, where N(v) is the open neighborhood of v. A set {f 1,f 2,?,f d } of total {k}-dominating functions on G with the property that $\sum_{i=1}^{d}f_{i}(v)\le k$ for each v?V(G), is called a total {k}-dominating family (of functions) on G. The maximum number of functions in a total {k}-dominating family on G is the total {k}-domatic number of G, denoted by $d_{t}^{\{k\}}(G)$ . Note that $d_{t}^{\{1\}}(G)$ is the classic total domatic number d t (G). In this paper we initiate the study of the total {k}-domatic number in graphs and we present some bounds for $d_{t}^{\{k\}}(G)$ . Many of the known bounds of d t (G) are immediate consequences of our results.

Signed total (k, k)-domatic number of a graph

Let G be a finite and simple graph with vertex set V (G), and let f: V(G) → {-1, 1} be a two-valued function. If k ≥ 1 is an integer and for each v ϵ V(G), where N(v) is the neighborhood of v, then f is a signed total k-dominating function on G. A set {f1, f2, …, fd} of signed total k-dominating functions on G with the property that for each x ϵ V (G), is called a signed total (k, k)-dominating family (of functions) on G. The maximum number of functions in a signed total (k, k)-dominating family on G is the signed total (k, k)-domatic number on G, denoted by dkst(G).In this paper we initiate the study of the signed total (k, k)-domatic number, and we present different bounds on dkst(G). Some of our results are extensions of known properties of the signed total domatic number dst(G) = d1st(G), given by Henning in 2006.

On the domatic and the total domatic numbers of the 2-section graph of the order-interval hypergraph of a finite poset

Let P be a finite poset and H ( P ) be the hypergraph whose vertices are the points of P and whose edges are the maximal intervals in P . We study the domatic number d ( G ( P ) ) and the total domatic number d t ( G ( P ) ) of the 2-section graph G ( P ) of H ( P ) . For the subset P l , u of P induced by consecutive levels ∪ i = l u N i of P , we give exact values of d ( G ( P l , u ) ) when P is the chain product C n 1 × C n 2 . According to the values of l , u , n 1 , n 2 , the maximal domatic partition is exhibited. Moreover, we give some exact values or lower bounds for d ( G ( P ∗ Q ) ) and d t ( G ( P l , u ) ) , when ∗ is the direct sum, the linear sum or the Cartesian product. Finally we show that the domatic number and the total domatic number problems in this class of graphs are NP-complete.

Signed total domatic number of a graph

Let G = (V, E) be a graph, and let f: V → {−1, 1} be a two-valued function. If \(\sum\limits_{x \in N(v)} {f(x) \geqslant 1} \) for each v e V, where N(v) is the open neighborhood of v, then f is a signed total dominating function on G. A set {f1, f2, …, fd} of signed total dominating functions on G with the property that \(\sum\limits_{i = 1}^d {f_i (x) \leqslant 1} \) for each x e V, is called a signed total dominating family (of functions) on G. The maximum number of functions in a signed total dominating family on G is the signed total domatic number on G, denoted by dts (G). The properties of the signed total domatic number dts (G) are studied in this paper. In particular, we give the sharp bounds of the signed total domatic number of regular graphs, complete bipartite graphs and complete graphs.

On the domatic and the total domatic numbers of the 2-section graph of the order-interval hypergraph of the finite poset

Abstract Let P be a finite poset and H ( P ) be the hypergraph whose vertices are the points of P and whose edges are the maximal intervals in P . We study the domatic and the total domatic numbers of the 2-section graph G ( P ) of H ( P ) . For the subset P l , u of P induced by consecutive levels ∪ i = l u N i of P , we give exact values of d ( G ( P l , u ) ) and maximal bound of d t ( G ( P l , u ) ) when P is the chain product C n 1 × C n 2 . Moreover, we give some exact values or lower bounds for d ( G ( P * Q ) ) and d t ( G ( P l , u ) ) , when * is either the direct sum or the linear sum. Finally we show that the domatic number and the total domatic number problems in this class of graphs are NP-complete.

Domination in Bipartite Graphs and in Their Complements

The domatic numbers of a graph G and of its complement \(\bar G\) were studied by J. E. Dunbar, T. W. Haynes and M. A. Henning. They suggested four open problems. We will solve the following ones:Characterize bipartite graphs G having \(d\left( G \right) = d\left( {\bar G} \right)\) Further, we will present a partial solution to the problem:Is it true that if G is a graph satisfying \(d\left( G \right) = d\left( {\bar G} \right)\) then \(\gamma \left( G \right) = \gamma \left( {\bar G} \right)\)?Finally, we prove an existence theorem concerning the total domatic number of a graph and of its complement

Domination in generalized Petersen graphs

Generalized Petersen graphs are certain graphs consisting of one quadratic factor. For these graphs some numerical invariants concerning the domination are studied, namely the domatic number \(d(G)\), the total domatic number \(d_t (G)\) and the \(k\)-ply domatic number \(d^k (G)\) for \(k = 2\) and \(k = 3\). Some exact values and some inequalities are stated.