Adel P . Kazemi
Uma, Mathematics, Faculty Member
Let G be a graph of order n and size m and let k≥ 1 be an integer. A k-tuple total dominating set in G is called a k-tuple total restrained dominating set of G if each vertex x∈ V(G)-S is adjacent to at least k vertices of V(G)-S. The... more
Let G be a graph of order n and size m and let k≥ 1 be an integer. A k-tuple total dominating set in G is called a k-tuple total restrained dominating set of G if each vertex x∈ V(G)-S is adjacent to at least k vertices of V(G)-S. The minimum number of vertices of a such sets in G are the k-tuple total restrained domination number γ_× k,t^r(G) of G. The maximum number of classes of a partition of V(G) such that its all classes are k-tuple total restrained dominating sets in G, is called the k-tuple total restrained domatic number of G. In this manuscript, we first find γ_× k,t^r(G), when G is complete graph, cycle, bipartite graph and the complement of path or cycle. Also we will find bounds for this number when G is a complete multipartite graph. Then we will know the structure of graphs G which γ_× k,t^r(G)=m, for some m≥ k+1 and give upper and lower bounds for γ_× k,t^r(G), when G is an arbitrary graph. Next, we mainly present basic properties of the k-tuple total restrained doma...
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The inflated graph G_I of a graph G with n(G) vertices is obtained from G by replacing every vertex of degree d of G by a clique, which is isomorph to the complete graph K_d, and each edge (x_i,x_j) of G is replaced by an edge (u,v) in... more
The inflated graph G_I of a graph G with n(G) vertices is obtained from G by replacing every vertex of degree d of G by a clique, which is isomorph to the complete graph K_d, and each edge (x_i,x_j) of G is replaced by an edge (u,v) in such a way that u∈ X_i, v∈ X_j, and two different edges of G are replaced by non-adjacent edges of G_I. For integer k≥ 1, the k-tuple total domination number γ_× k,t(G) of G is the minimum cardinality of a k-tuple total dominating set of G, which is a set of vertices in G such that every vertex of G is adjacent to at least k vertices in it. For existing this number, must the minimum degree of G is at least k. Here, we study the k-tuple total domination number in inflated graphs when k≥ 2. First we prove that n(G)k≤γ_× k,t(G_I)≤ n(G)(k+1)-1, and then we characterize graphs G that the k-tuple total domination number number of G_I is n(G)k or n(G)k+1. Then we find bounds for this number in the inflated graph G_I, when G has a cut-edge e or cut-vertex v, ...
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Let G=(V,E) be a simple graph. For any integer k≥ 1, a subset of V is called a k-tuple total dominating set of G if every vertex in V has at least k neighbors in the set. The minimum cardinality of a minimal k-tuple total dominating set... more
Let G=(V,E) be a simple graph. For any integer k≥ 1, a subset of V is called a k-tuple total dominating set of G if every vertex in V has at least k neighbors in the set. The minimum cardinality of a minimal k-tuple total dominating set of G is called the k-tuple total domination number of G. In this paper, we introduce the concept of upper k-tuple total domination number of G as the maximum cardinality of a minimal k-tuple total dominating set of G, and study the problem of finding a minimal k-tuple total dominating set of maximum cardinality on several classes of graphs, as well as finding general bounds and characterizations. Also, we find some results on the upper k-tuple total domination number of the Cartesian and cross product graphs.
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The inflation GI of a graph G with n(G) vertices and m(G) edges is obtained from G by replacing every vertex of degree d of G by a clique, which is isomorphic to the complete graph K d , and each edge (xi, xj) of G is replaced by an edge... more
The inflation GI of a graph G with n(G) vertices and m(G) edges is obtained from G by replacing every vertex of degree d of G by a clique, which is isomorphic to the complete graph K d , and each edge (xi, xj) of G is replaced by an edge (u, v) in such a way that u ∈ Xi, v ∈ Xj, and two different edges of G are replaced by non-adjacent edges of GI. The total domination number γt(G) of a graph G is the minimum cardinality of a total dominating set, which is a set of vertices such that every vertex of G is adjacent to one vertex of it. A graph is Kr-covered if every vertex of it is contained in a clique Kr. Cockayne et al. in [Total domination in Kr-covered graphs, Ars Combin. 71 (2004) 289-303] conjectured that the total domination number of every Kr-covered graph with n vertices and no Kr-component is at most 2n r+1. This conjecture has been proved only for 3 ≤ r ≤ 6. In this paper, we prove this conjecture for a big family of Kr-covered graphs. 1. Preliminaries Let G = (V, E) be a ...
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Let G be a graph with minimum degree at least 2. A set D⊆ V is a double total dominating set of G if each vertex is adjacent to at least two vertices in D. The double total domination number γ _× 2,t(G) of G is the minimum cardinality of... more
Let G be a graph with minimum degree at least 2. A set D⊆ V is a double total dominating set of G if each vertex is adjacent to at least two vertices in D. The double total domination number γ _× 2,t(G) of G is the minimum cardinality of a double total dominating set of G. In this paper, we will find double total domination number of Harary graphs.
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A k-tuple total dominating set (kTDS) of a graph G is a set S of vertices in which every vertex in G is adjacent to at least k vertices in S; the minimum size of a kTDS is denoted ?xk,t(G). We give a Vizing-like inequality for Cartesian... more
A k-tuple total dominating set (kTDS) of a graph G is a set S of vertices in which every vertex in G is adjacent to at least k vertices in S; the minimum size of a kTDS is denoted ?xk,t(G). We give a Vizing-like inequality for Cartesian product graphs, namely ?xk,t(G) ?xk,t(H)? 2k?xk,t(G_H) provided ?xk,t(G) ? 2k?(G) holds, where ? denotes the packing number. We also give bounds on ?xk,t(G_H) in terms of (open) packing numbers, and consider the extremal case of ?xk,t(Kn_Km), i.e., the rook?s graph, giving a constructive proof of a general formula for ?x2,t(Kn_Km).
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Abstract In this paper, we consider an infinite sub-family of the generalized Petersen graphs P(n, k), for n = 2k + 1 ≥ 3. A. Behzad et al. in [1] gave the upper bound ⌈3n/5⌉ for the domination number of this kind of graphs, where n ≥ 3.... more
Abstract In this paper, we consider an infinite sub-family of the generalized Petersen graphs P(n, k), for n = 2k + 1 ≥ 3. A. Behzad et al. in [1] gave the upper bound ⌈3n/5⌉ for the domination number of this kind of graphs, where n ≥ 3. They conjectured that the upper bound is sharp. In this paper, we first find a lower bound for the domination number of P(n, k) graphs, where n ≥ 7, and then we give some of the this kind of graphs that their domination numbers achieve the upper bound ⌈3n/5⌉. Also we present their independent domination numbers.
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Given a graph $G$, the total dominator coloring problem seeks a proper coloring of $G$ with the additional property that every vertex in the graph is adjacent to all vertices of a color class. We seek to minimize the number of color... more
Given a graph $G$, the total dominator coloring problem seeks a proper coloring of $G$ with the additional property that every vertex in the graph is adjacent to all vertices of a color class. We seek to minimize the number of color classes. We initiate to study this problem on several classes of graphs, as well as finding general bounds and characterizations. We also compare the total dominator chromatic number of a graph with the chromatic number and the total domination number of it.
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Let k be a positive integer. A Roman k-dominating function on a graph G is a labeling f:V (G)⟶{0, 1, 2} such that every vertex with label 0 has at least k neighbors with label 2. A set {f_1,f_2,...,f_d} of distinct Roman k-dominating... more
Let k be a positive integer. A Roman k-dominating function on a graph G is a labeling f:V (G)⟶{0, 1, 2} such that every vertex with label 0 has at least k neighbors with label 2. A set {f_1,f_2,...,f_d} of distinct Roman k-dominating functions on G with the property that ∑_i=1^df_i(v)< 2k for each v∈ V(G), is called a Roman (k,k)-dominating family (of functions) on G. The maximum number of functions in a Roman (k,k)-dominating family on G is the Roman (k,k)-domatic number of G, denoted by d_R^k(G). Note that the Roman (1,1)-domatic number d_R^1(G) is the usual Roman domatic number d_R(G). In this paper we initiate the study of the Roman (k,k)-domatic number in graphs and we present sharp bounds for d_R^k(G). In addition, we determine the Roman (k,k)-domatic number of some graphs. Some of our results extend those given by Sheikholeslami and Volkmann in 2010 for the Roman domatic number.
The total dominator total coloring of a graph is a total coloring of the graph such that each object of the graph is adjacent or incident to every object of some color class. The minimum namber of the color classes of a total dominator... more
The total dominator total coloring of a graph is a total coloring of the graph such that each object of the graph is adjacent or incident to every object of some color class. The minimum namber of the color classes of a total dominator total coloring of a graph is called the total dominator total chromatic number of the graph. Here, we will find the total dominator chromatic numbers of cycles and paths.
Total domination number of a graph without isolated vertex is the minimum cardinality of a total dominating set, that is, a set of vertices such that every vertex of the graph is adjacent to at least one vertex of the set. Henning and... more
Total domination number of a graph without isolated vertex is the minimum cardinality of a total dominating set, that is, a set of vertices such that every vertex of the graph is adjacent to at least one vertex of the set. Henning and Kazemi in [4] extended this definition as follows: for any positive integer k, and any graph G with minimum degree-k, a set D of vertices is a k-tuple total dominating set of G if each vertex of G is adjacent to at least k vertices in D. The k-tuple total domination number γ×k,t(G) of G is the minimum cardinality of a k-tuple total dominating set of G. In this paper, we give some upper bounds for the k-tuple total domination number of the supergeneralized Petersen graphs. Also we calculate the exact amount of this number for some of them.
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Let G = (V,E) be a graph. A k-coloring of a graph G is a labeling f: V (G) → T, where | T | = k and it is proper if the adjacent vertices have different labels. A graph is k-colorable if it has a proper k-coloring. The chromatic number... more
Let G = (V,E) be a graph. A k-coloring of a graph G is a labeling f: V (G) → T, where | T | = k and it is proper if the adjacent vertices have different labels. A graph is k-colorable if it has a proper k-coloring. The chromatic number χ(G) is the least k such that G is k-colorable. Here we study chromatic numbers in some kinds of Harary graphs.
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Let k be a positive integer and let G = (V,E) be a simple graph. The k-tuple domination number γ×k(G) of G is the minimum cardinality of a k-tuple dominating set S, a set that for every vertex v ∈ V, |NG[v]∩S | ≥ k. Also the total... more
Let k be a positive integer and let G = (V,E) be a simple graph. The k-tuple domination number γ×k(G) of G is the minimum cardinality of a k-tuple dominating set S, a set that for every vertex v ∈ V, |NG[v]∩S | ≥ k. Also the total k-domination number γ×k,t(G) of G is the minimum cardi-nality of a total k-dominating set S, a set that for every vertex v ∈ V, |NG(v)∩S | ≥ k. The k-transversal number τk(H) of a hypergraph H is the minimum size of a subset S ⊆ V (H) such that |S ∩ e | ≥ k for every edge e ∈ E(H). We know that for any graph G of order n with minimum degree at least k, γ×k(G) ≤ γ×k,t(G) ≤ n. Obviously for every k-regular graph, the upper bound n is sharp. Here, we give a sufficient condition for γ×k,t(G) < n. Then we characterize complete multipartite graphs G with γ×k(G) = γ×k,t(G). We also state that the total k-domination number of a graph is the k-transversal number of its open neighborhood hypergraph, and also the domination number of a graph is the transversal num...
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Given a graph G, the total dominator coloring problem seeks a proper coloring of G with the additional property that every vertex in the graph is adjacent to all vertices of a color class. We seek to minimize the number of color classes.... more
Given a graph G, the total dominator coloring problem seeks a proper coloring of G with the additional property that every vertex in the graph is adjacent to all vertices of a color class. We seek to minimize the number of color classes. We initiate to study this problem on several classes of graphs, as well as finding general bounds and characterizations. We also compare the total dominator chromatic number of a graph with the chromatic number and the total domination number of it.
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For any integer k ≥ 1 and any graph G = (V,E) with minimum degree at least k−1, we define a function f : V → {0, 1, 2} as a Roman k-tuple dominating function on G if for any vertex v with f(v) = 0 there exist at least k and for any vertex... more
For any integer k ≥ 1 and any graph G = (V,E) with minimum degree at least k−1, we define a function f : V → {0, 1, 2} as a Roman k-tuple dominating function on G if for any vertex v with f(v) = 0 there exist at least k and for any vertex v with f(v) 6= 0 at least k − 1 vertices in its neighborhood with f(w) = 2. The minimum weight of a Roman k-tuple dominating function f on G is called the Roman k-tuple domination number of the graph where the weight of f is f(V ) = ∑ v∈V f(v). In this paper, we initiate to study the Roman k-tuple domination number of a graph, by giving some sharp bounds for the Roman k-tuple domination number of a garph, the Mycieleskian of a graph, and the corona graphs. Also finding the Roman k-tuple domination number of some known graphs is our other goal. Some of our results extend these one given by Cockayne and et al. [1] in 2004 for the Roman domination number.
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For any integer $k\geq 1$ and any graph $G=(V,E)$ with minimum degree at least $k-1$, we define a function $f:V\rightarrow \{0,1,2\}$ as a Roman $k$-tuple dominating function on $G$ if for any vertex $v$ with $f(v)=0$ there exist at least... more
For any integer $k\geq 1$ and any graph $G=(V,E)$ with minimum degree at least $k-1$, we define a function $f:V\rightarrow \{0,1,2\}$ as a Roman $k$-tuple dominating function on $G$ if for any vertex $v$ with $f(v)=0$ there exist at least $k$ and for any vertex $v$ with $f(v)\neq 0$ at least $k-1$ vertices in its neighborhood with $f(w)=2$. The minimum weight of a Roman $k$-tuple dominating function $f$ on $G$ is called the Roman $k$-tuple domination number of the graph where the weight of $f$ is $f(V)=\sum_{v\in V}f(v)$. In this paper, we initiate to study the Roman $k$-tuple domination number of a graph, by giving some sharp bounds for the Roman $k$-tuple domination number of a garph, the Mycieleskian of a graph, and the corona graphs. Also finding the Roman $k$-tuple domination number of some known graphs is our other goal. Some of our results extend these one given by Cockayne and et al.} in 2004 for the Roman domination number.
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For every positive integer k, a set S of vertices in a graph G = (V;E) is a k- tuple dominating set of G if every vertex of V -S is adjacent to at least k vertices and every vertex of S is adjacent to at least k - 1 vertices in S. The... more
For every positive integer k, a set S of vertices in a graph G = (V;E) is a k- tuple dominating set of G if every vertex of V -S is adjacent to at least k vertices and every vertex of S is adjacent to at least k - 1 vertices in S. The minimum cardinality of a k-tuple dominating set of G is the k-tuple domination number of G. When k = 1, a k-tuple domination number is the well-studied domination number. We define the k-tuple domatic number of G as the largest number of sets in a partition of V into k-tuple dominating sets. Recall that when k = 1, a k-tuple domatic number is the well-studied domatic number. In this work, we derive basic properties and bounds for the k-tuple domatic number.
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For a graph $G=(V,E)$, we call a subset $ S\subseteq V \cup E$ a total mixed dominating set of $G$ if each element of $V \cup E$ is either adjacent or incident to an element of $S$, and the total mixed domination number $\gamma_{tm}(G)$... more
For a graph $G=(V,E)$, we call a subset $ S\subseteq V \cup E$ a total mixed dominating set of $G$ if each element of $V \cup E$ is either adjacent or incident to an element of $S$, and the total mixed domination number $\gamma_{tm}(G)$ of $G$ is the minimum cardinality of a total mixed dominating set of $G$. In this paper, we initiate to study the total mixed domination number of a connected graph by giving some tight bounds in terms of some parameters such as order and total domination numbers of the graph and its line graph. Then we discuss on the relation between total mixed domination number of a graph and its diameter. Studing of this number in trees is our next work. Also we show that the total mixed domination number of a graph is equale to the total domination number of a graph which is obtained by the graph. Giving the total mixed domination numbers of some special graphs is our last work.
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For any integer k 1, a set S of vertices in a graph G = (V;E) is a k-tuple total dominating set of G if any vertex of G is adjacent to at least k vertices in S, and any vertex of V S is adjacent to at least k vertices in V S. The minimum... more
For any integer k 1, a set S of vertices in a graph G = (V;E) is a k-tuple total dominating set of G if any vertex of G is adjacent to at least k vertices in S, and any vertex of V S is adjacent to at least k vertices in V S. The minimum number of vertices of such a set in G we call the k-tuple total restrained domination number of G. The maximum number of classes of a partition of V such that its all classes are k-tuple total restrained dominating sets in G we call the k-tuple total restrained domatic number of G. In this paper, we give some sharp bounds for the k-tuple total restrained domination number of a graph, and also calculate it for some of the known graphs. Next, we mainly present basic proper- ties of the k-tuple total restrained domatic number of a graph. Keywords: k-tuple total domination number, k-tuple total do- matic number, k-tuple total restrained domination number, k-tuple total restrained domatic number. MSC(2010): Primary: 05C69.
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For every positive integer $k$, a set $S$ of vertices in a graph $G=(V,E)$ is a $k$-tuple dominating set of $G$ if every vertex of $V-S$ is adjacent to least $k$ vertices and every vertex of $S$ is adjacent to least $k-1$ vertices in $S$.... more
For every positive integer $k$, a set $S$ of vertices in a graph $G=(V,E)$ is a $k$-tuple dominating set of $G$ if every vertex of $V-S$ is adjacent to least $k$ vertices and every vertex of $S$ is adjacent to least $k-1$ vertices in $S$. The minimum cardinality of a $k$-tuple dominating set of $G$ is the $k$-tuple domination number of $G$. When $k=1$, a $k$-tuple domination number is the well-studied domination number. We define the $k$-tuple domatic number of $G$ as the largest number of sets in a partition of $V$ into $k$-tuple dominating sets. Recall that when $k=1$, a $k$-tuple domatic number is the well-studied domatic number. In this work, we derive basic properties and bounds for the $k$-tuple domatic number.
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For k ≥ 1 an integer, a set S of vertices in a graph G with minimum degree at least k − 1 is a k-tuple dominating set of G if every vertex of S is adjacent to at least k−1 vertices in S and every vertex of V (G)\S is adjacent to at least... more
For k ≥ 1 an integer, a set S of vertices in a graph G with minimum degree at least k − 1 is a k-tuple dominating set of G if every vertex of S is adjacent to at least k−1 vertices in S and every vertex of V (G)\S is adjacent to at least k vertices in S; that is, |NG[v] ∩ S| ≥ k for every vertex v of G where NG[v] denotes the closed neighborhood of v which consists of v and all neighbors of v. A k-tuple restrained dominating set of G is a k-tuple dominating set S of G with the additional property that every vertex outside S has at least k neighbors outside S. The minimum cardinality of a k-tuple restrained dominating set of G is the k-tuple restrained domination number of G. When k = 1, the k-tuple restrained domination number is the well-studied restrained domination number. In this paper, we determine the k-tuple restrained domination number of several classes of graphs. Tight upper bounds on the k-tuple restrained domination number of a general graph are established. We present b...
In a graph G with δ(G) ≥ k ≥ 1, a k-tuple total restrained dominating set S is a subset of V(G) such that each vertex of V(G) is adjacent to at least k vertices of S and also each vertex ofV(G) − S is adjacent to at least k vertices of... more
In a graph G with δ(G) ≥ k ≥ 1, a k-tuple total restrained dominating set S is a subset of V(G) such that each vertex of V(G) is adjacent to at least k vertices of S and also each vertex ofV(G) − S is adjacent to at least k vertices of V(G) − S.Theminimumnumber of vertices of such sets inG is the k-tuple total restrained domination number ofG. In [k-tuple total restrained domination/domatic in graphs, BIMS], the author initiated the study of the k-tuple total restrained domination number in graphs. In this paper, we continue it in the complementary prism of a graph.
Given a graph $G$, the total dominator coloring problem seeks a proper coloring of $G$ with the additional property that every vertex in the graph is adjacent to all vertices of a color class. We seek to minimize the number of... more
Given a graph $G$, the total dominator coloring problem seeks a proper coloring of $G$ with the additional property that every vertex in the graph is adjacent to all vertices of a color class. We seek to minimize the number of color classes. We initiate to study this problem on several classes of graphs, as well as finding general bounds and characterizations. We also compare the total dominator chromatic number of a graph with the chromatic number and the total domination number of it.
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Let k be a positive integer. A subset S of V (G) in a graph G is a k-tuple total dominating set of G if every vertex of G has at least k neighbors in S. The k-tuple total domination number γ×k,t(G) of G is the minimum cardinality of a... more
Let k be a positive integer. A subset S of V (G) in a graph G is a k-tuple total dominating set of G if every vertex of G has at least k neighbors in S. The k-tuple total domination number γ×k,t(G) of G is the minimum cardinality of a k-tuple total dominating set of G. In this paper for a given graph G with minimum degree at least k, we find some sharp lower and upper bounds on the k-tuple total domination number of the m-Mycieleskian graph μm(G) of G in terms on k and γ×k,t(G). Specially we give the sharp bounds γ×k,t(G) + 1 and γ×k,t(G) + k for γ×k,t(μ1(G)), and characterize graphs with γ×k,t(μ1(G)) = γ×k,t(G) + 1.
Let $S$ be a set of vertices of a graph $G$. Let $cl(S)$ be the set of vertices built from $S$, by iteratively applying the following propagation rule: if a vertex and all but exactly one of its neighbors are in $cl(S)$, then the... more
Let $S$ be a set of vertices of a graph $G$. Let $cl(S)$ be the set of vertices built from $S$, by iteratively applying the following propagation rule: if a vertex and all but exactly one of its neighbors are in $cl(S)$, then the remaining neighbor is also in $cl(S)$. A set $S$ is called a zero forcing set of $G$ if $cl(S)=V(G)$. The zero forcing number $Z(G)$ of $G$ is the minimum cardinality of a zero forcing set. Let $cl(N[S])$ be the set of vertices built from the closed neighborhood $N[S]$ of $S$, by iteratively applying the previous propagation rule. A set $S$ is called a power dominating set of $G$ if $cl(N[S])=V(G)$. The power domination number $\gp(G)$ of $G$ is the minimum cardinality of a power dominating set. In this paper, we characterize the set of all graphs $G$ for which $Z(G)=2$. On the other hand, we present a variety of sufficient and/or necessary conditions for a graph $G$ to satisfy $1 \le \gp(G) \le 2$.
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A total dominator coloring of a graph $G$ is a proper coloring of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number $\chi_d^t(G)$ of $G$ is the minimum number of... more
A total dominator coloring of a graph $G$ is a proper coloring of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number $\chi_d^t(G)$ of $G$ is the minimum number of color classes in a total dominator coloring of it. In [Total dominator chromatic number in graphs, Transactions on Combinatorics Vol. 4 No. 2 (2015), pp. 57--68] the author initiated to study this number in graphs. Here, we continue the studying of it on central graphs.
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The circulant graph $C_n(S)$ with connection set $S\subseteq \{1,2,\cdots,n\}$ is the graph with vertex set $V=\{1,\ldots, n\}$ and two vertices $x,y$ are adjacent if $|x-y|\in S$. In this paper, we will calculate the total dominator... more
The circulant graph $C_n(S)$ with connection set $S\subseteq \{1,2,\cdots,n\}$ is the graph with vertex set $V=\{1,\ldots, n\}$ and two vertices $x,y$ are adjacent if $|x-y|\in S$. In this paper, we will calculate the total dominator chromatic number of the circulant graph $C_n(\{a,b\})$ when $n\geq 6$, $gcd(a,n)=1$ and $ a^{-1}b\equiv 3 \pmod{n}$.
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Let k be a positive integer, and let G be a graph with minimum degree at least k. In their study 2010 , Henning and Kazemi defined the k-tuple total domination number γ×k,t G of G as the minimum cardinality of a k-tuple total dominating... more
Let k be a positive integer, and let G be a graph with minimum degree at least k. In their study 2010 , Henning and Kazemi defined the k-tuple total domination number γ×k,t G of G as the minimum cardinality of a k-tuple total dominating set of G, which is a vertex set such that every vertex of G is adjacent to at least k vertices in it. If G is the complement of G, the complementary prism GG of G is the graph formed from the disjoint union of G and G by adding the edges of a perfect matching between the corresponding vertices of G and G. In this paper, we extend some of the results of Haynes et al. 2009 for the k-tuple total domination number and also obtain some other new results. Also we find the k-tuple total domination number of the complementary prism of a cycle, a path, or a complete multipartite graph.
Let G = (V,E) be a simple graph. For any integer k ≥ 1, a subset of V is called a k-tuple total dominating set of G if every vertex in V has at least k neighbors in the set. The minimum cardinality of a minimal k-tuple total dominating... more
Let G = (V,E) be a simple graph. For any integer k ≥ 1, a subset of V is called a k-tuple total dominating set of G if every vertex in V has at least k neighbors in the set. The minimum cardinality of a minimal k-tuple total dominating set of G is called the k-tuple total domination number of G. In this paper, we introduce the concept of upper k-tuple total domination number of G as the maximum cardinality of a minimal k-tuple total dominating set of G, and study the problem of finding a minimal k-tuple total dominating set of maximum cardinality on several classes of graphs, as well as finding general bounds and characterizations. Also, we find some results on the upper k-tuple total domination number of the Cartesian and cross product graphs.
Total dominator total coloring of a graph is a total coloring of the graph such that each object of the graph is adjacent or incident to every object of some color class. The minimum namber of the color classes of a total dominator total... more
Total dominator total coloring of a graph is a total coloring of the graph such that each object of the graph is adjacent or incident to every object of some color class. The minimum namber of the color classes of a total dominator total coloring of a graph is called the total dominator total chromatic number of the graph. Here, we will find the total dominator chromatic numbers of wheels, complete bipartite graphs and complete graphs.
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Let $G$ be a graph of order $n$ and size $m$ and let $k\geq 1$ be an integer. A $k$-tuple total dominating set in $G$ is called a $k$-tuple total restrained dominating set of $G$ if each vertex $x\in V(G)-S$ is adjacent to at least $k$... more
Let $G$ be a graph of order $n$ and size $m$ and let $k\geq 1$ be an integer. A $k$-tuple total dominating set in $G$ is called a $k$-tuple total restrained dominating set of $G$ if each vertex $x\in V(G)-S$ is adjacent to at least $k$ vertices of $V(G)-S$. The minimum number of vertices of a such sets in $G$ are the $k$-tuple total restrained domination number $\gamma_{\times k,t}^{r}(G)$ of $G$. The maximum number of classes of a partition of $V(G)$ such that its all classes are $k$-tuple total restrained dominating sets in $G$, is called the $k$-tuple total restrained domatic number of $G$. In this manuscript, we first find $\gamma_{\times k,t}^{r}(G)$, when $G$ is complete graph, cycle, bipartite graph and the complement of path or cycle. Also we will find bounds for this number when $G$ is a complete multipartite graph. Then we will know the structure of graphs $G$ which $\gamma_{\times k,t}^{r}(G)=m$, for some $m\geq k+1$ and give upper and lower bounds for $\gamma_{\times k,t...
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A total dominator coloring of a graph G is a proper coloring of G in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic numbert (G) of G is the minimum number of color classes in... more
A total dominator coloring of a graph G is a proper coloring of G in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic numbert (G) of G is the minimum number of color classes in a total dominator coloring of it. In (Total dominator chromatic numer in graphs, submitted) the author initialed to study this number in graphs and obtained some important results. Here, we continue it in Mycieleskian graphs. We show that the total dominator chromatic number of the Mycieleskian of a graph G belongs to betweent (G) + 1 andt (G) + 2, and then characterize the family of graphs the their total dominator chromatic numbers are each of them.
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For a graph $G=(V,E)$, a function $f:V\rightarrow \{0,1,2\}$ is called Roman dominating function (RDF) if for any vertex $v$ with $f(v)=0$, there is at least one vertex $w$ in its neighborhood with $f(w)=2$. The weight of an RDF $f$ of... more
For a graph $G=(V,E)$, a function $f:V\rightarrow \{0,1,2\}$ is called Roman dominating function (RDF) if for any vertex $v$ with $f(v)=0$, there is at least one vertex $w$ in its neighborhood with $f(w)=2$. The weight of an RDF $f$ of $G$ is the value $f(V)=\sum_{v\in V}f(v)$. The minimum weight of an RDF of $G$ is its Roman domination number and denoted by $\gamma_ R(G)$. In this paper, we first show that $\gamma_{R}(G)+1\leq \gamma_{R}(\mu (G))\leq \gamma_{R}(G)+2$, where $\mu (G)$ is the Mycielekian graph of $G$, and then characterize the graphs achieving equality in these bounds. Then for any positive integer $m$, we compute the Roman domination number of the $m$-Mycieleskian $\mu_{m}(G)$ of a special Roman graph $G$ in terms on $\gamma_R(G)$. Finally we present several graphs to illustrate the discussed graphs.
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Total domination number of a graph without isolated vertex is the minimum cardinality of a total dominating set, that is, a set of vertices such that every vertex of the graph is adjacent to at least one vertex of the set. Henning and... more
Total domination number of a graph without isolated vertex is the minimum cardinality of a total dominating set, that is, a set of vertices such that every vertex of the graph is adjacent to at least one vertex of the set. Henning and Kazemi in [4] extended this definition as follows: for any positive integer $k$, and any graph $G$ with minimum degree-$k$, a set $D $ of vertices is a $k$-tuple total dominating set of $G$ if each vertex of $G$ is adjacent to at least $k$ vertices in $D$. The $k$-tuple total domination number $\gamma _{\times k,t}(G)$ of $G$ is the minimum cardinality of a $k$-tuple total dominating set of $G$. In this paper, we give some upper bounds for the $k$-tuple total domination number of the supergeneralized Petersen graphs. Also we calculate the exact amount of this number for some of them.
Let G be a graph with the vertex set V (G) and S be a subset of V (G). Let cl(S) be the set of vertices built from S, by iteratively applying the following propagation rule: if a vertex and all of its neighbors except one of them are in... more
Let G be a graph with the vertex set V (G) and S be a subset of V (G). Let cl(S) be the set of vertices built from S, by iteratively applying the following propagation rule: if a vertex and all of its neighbors except one of them are in cl(S), then the exceptional neighbor is also in cl(S). A set S is called a zero forcing set of G if cl(S) = V (G). The zero forcing number Z(G) of G is the minimum cardinality of a zero forcing set. Let cl(N [S]) be the set of vertices built from the closed neighborhood N [S] of S, by iteratively applying the previous propagation rule. A set S is called a power dominating set of G if cl(N [S]) = V (G). The power domination number γp(G) of G is the minimum cardinality of a power dominating set. In this paper, we present some families of graphs that their power domination number is 1 or 2.
The most famous open problem involving domination in graphs is Vizing's conjecture which states the domination number of the Cartesian product of any two graphs is at least as large as the product of their domination numbers. In this... more
The most famous open problem involving domination in graphs is Vizing's conjecture which states the domination number of the Cartesian product of any two graphs is at least as large as the product of their domination numbers. In this paper, we investigate a similar problem for the k-tuple total domination number. Then we calculate the k-tuple total domination number of the Cartesian product of two complete graphs.
Given a graph $G$, the total dominator coloring problem seeks a proper coloring of $G$ with the additional property that every vertex in the graph is adjacent to all vertices of a color class. We seek to minimize the number of color... more
Given a graph $G$, the total dominator coloring problem seeks a proper coloring of $G$ with the additional property that every vertex in the graph is adjacent to all vertices of a color class. We seek to minimize the number of color classes. We study this problem on several classes of graphs, as well as finding general bounds and characterizations. We also show the relation between total dominator chromatic number and chromatic number and total domination number.
Let $G$ be a graph with minimum degree at least 2. A set $D\subseteq V$ is a double total dominating set of $G$ if each vertex is adjacent to at least two vertices in $D$. The double total domination number $\gamma _{\times 2,t}(G)$ of... more
Let $G$ be a graph with minimum degree at least 2. A set $D\subseteq V$ is a double total dominating set of $G$ if each vertex is adjacent to at least two vertices in $D$. The double total domination number $\gamma _{\times 2,t}(G)$ of $G$ is the minimum cardinality of a double total dominating set of $G$. In this paper, we will find double total domination number of Harary graphs.
In this paper, we first present the relation between a transversal in a Latin square with some concepts in its Latin square graph, and give an equivalent condition for a Latin square has an orthogonal mate. The most famous open problem... more
In this paper, we first present the relation between a transversal in a Latin square with some concepts in its Latin square graph, and give an equivalent condition for a Latin square has an orthogonal mate. The most famous open problem involving Combinatorics is to find maximum number of disjoint transversals in a Latin square. So finding some family of decomposable Latin squares into disjoint transversals is our next aim. In the next section, we give an equivalent statement of a conjecture which has been attributed to Brualdi, Stein and Ryser by the concept of quasi-transversal. Finally, we prove the truth of the Rodney's conjecture for a family of graphs.
The total dominator total coloring of a graph is a total coloring of the graph such that each object of the graph is adjacent or incident to every object of some color class. The minimum namber of the color classes of a total dominator... more
The total dominator total coloring of a graph is a total coloring of the graph such that each object of the graph is adjacent or incident to every object of some color class. The minimum namber of the color classes of a total dominator total coloring of a graph is called the total dominator total chromatic number of the graph. Here, we will find the total dominator chromatic numbers of cycles and paths.
The inflated graph GI of a graph G with n(G) vertices is obtained from G by replacing every vertex of degree d of G by a clique, which is isomorph to the complete graph Kd, and each edge (xi, xj) of G is replaced by an edge (u, v) in such... more
The inflated graph GI of a graph G with n(G) vertices is obtained from G by replacing every vertex of degree d of G by a clique, which is isomorph to the complete graph Kd, and each edge (xi, xj) of G is replaced by an edge (u, v) in such a way that u ∈ Xi, v ∈ Xj , and two different edges of G are replaced by non-adjacent edges of GI . For integer k ≥ 1, the k-tuple total domination number γ ×k,t(G) of G is the minimum cardinality of a k-tuple total dominating set of G, which is a set of vertices in G such that every vertex of G is adjacent to at least k vertices in it. For existing this number, must the minimum degree of G is at least k. Here, we study the k-tuple total domination number in inflated graphs when k ≥ 2. First we prove that n(G)k ≤ γ ×k,t(GI ) ≤ n(G)(k + 1) − 1, and then we characterize graphs G that the k-tuple total domination number number of GI is n(G)k or n(G)k + 1. Then we find bounds for this number in the inflated graph GI , when G has a cut-edge e or cut-ver...
For a graph G = (V,E), we call a subset S ⊆ V ∪ E a total mixed dominating set of G if each element of V ∪ E is either adjacent or incident to an element of S, and the total mixed domination number γtm(G) of G is the minimum cardinality... more
For a graph G = (V,E), we call a subset S ⊆ V ∪ E a total mixed dominating set of G if each element of V ∪ E is either adjacent or incident to an element of S, and the total mixed domination number γtm(G) of G is the minimum cardinality of a total mixed dominating set of G. In this paper, we initiate to study the total mixed domination number of a connected graph by giving some tight bounds in terms of some parameters such as order and total domination numbers of the graph and its line graph. Then we discuss on the relation between total mixed domination number of a graph and its diameter. Studing of this number in trees is our next work. Also we show that the total mixed domination number of a graph is equale to the total domination number of a graph which is obtained by the graph. Giving the total mixed domination numbers of some special graphs is our last work.