Total Domination In Graphs Research Articles

Total Domination in Graphs: New Perspectives and Results

In this research paper, we initiate a study of such questions in the present context of total domination theory within graph theory from new perspectives and results concerned with formal definitions, mathematical formulations, theorems and practical applications. Description Total domination is a tool used to determine the graph that has minimum number of vertices can monitor or control all other vertices - factors of interest for problems related, amongst others fields as network optimization and resilience analysis. This paper covers fundamental definitions as well their most recent theoretical bounds, algorithmic complexities and practical examples arising in applications from the real world. The obtained results could be helpful in deciphering, exploiting and administering the concept of total domination further.

Proper total domination in graphs

Proper total domination in graphs

Equality of total domination and chromatic total domination in graphs

Let be a simple, finite and undirected graph and without isolated vertex. A subset D of V is said to be dominating set if for every in there exist a vertex in such that and are adjacent. The minimum cardinality of a dominating set of is called the domination number of and is denoted by . is minimal dominating set of a graph if no proper subset of is a dominating set of . is a total dominating set of if has no isolates. The minimum cardinality of a total dominating set of is called the total domination number of and is denoted by . A Subset D of V is said to be a chromatic total dominating set if D is a total dominating set and =. The minimum cardinality of the chromatic total dominating set is called a chromatic total dominating number .In any graph G ,every chromatic total dominating set is a total dominating set.But converse is not true.In some graphs ,every set is a chromatic total dominating set.

On k-Fair Total Domination in Graphs

Let G = (V (G), E(G)) be a simple non-empty graph. For an integer k ≥ 1, a k-fairtotal dominating set (kf td-set) is a total dominating set S ⊆ V (G) such that |NG(u) ∩ S| = k for every u ∈ V (G)\S. The k-fair total domination number of G, denoted by γkf td(G), is the minimum cardinality of a kf td-set. A k-fair total dominating set of cardinality γkf td(G) is called a minimum k-fair total dominating set or a γkf td-set. We investigate the notion of k-fair total domination in this paper. We also characterize the k-fair total dominating sets in the join, corona, lexicographic product and Cartesian product of graphs and determine the exact values or sharpbounds of their corresponding k-fair total domination number.

Total and Secure Domination for Corona Product of Two Fuzzy Soft Graphs

Fuzzy sets and soft sets are two different soft computing models for representing vagueness and uncertainty. On the other domination is a rapidly developing area of research in graph theory, and its various applications to ad hoc networks, distributed computing, social networks and web graphs partly explain the increased interest. This concept was introduced by [Benecke S, Cockayne EJ, Mynhardt CM. Secure total domination in graphs. Util Math. 2007;74:247–259.] in 2007 and Go and Canoy continue the study of these notions [Canoy RS, Go CE. Domination in the corona and join of graphs. Int Math Forum. 2011;6(16):763–771.] and afterward introduce total dominating and secure total dominating sets. Also graph operations like corona product play a very important role in mathematical chemistry, since some chemically interesting graphs can be obtained from some simpler graphs by different graph operations. In this paper, we characterised the dominating, total dominating, and secure total dominating sets in the corona of two fuzzy soft connected.

Algorithm and hardness results on neighborhood total domination in graphs

A set D⊆V of a graph G=(V,E) is called a neighborhood total dominating set of G if D is a dominating set and the subgraph of G induced by the open neighborhood of D has no isolated vertex. Given a graph G, Min-NTDS is the problem of finding a neighborhood total dominating set of G of minimum cardinality. The decision version of Min-NTDS is known to be NP-complete for bipartite graphs and chordal graphs. In this paper, we extend this NP-completeness result to undirected path graphs, chordal bipartite graphs, and planar graphs. We also present a linear time algorithm for computing a minimum neighborhood total dominating set in proper interval graphs. We show that for a given graph G=(V,E), Min-NTDS cannot be approximated within a factor of (1−ε)log⁡|V|, unless NP⊆DTIME(|V|O(log⁡log⁡|V|)) and can be approximated within a factor of O(log⁡Δ), where Δ is the maximum degree of the graph G. Finally, we show that Min-NTDS is APX-complete for graphs of degree at most 3.

Chromatic total domination in graphs

Let G = (V, E) be a simple, finite and undirected graph and without isolated vertex. A set D ⊆ V is said to be chromatic total dominating set of G if D is a total dominating set and χ(<D>) = χ(G). The minimum cardinality of a chromatic total dominating set of G is called the chromatic total domination number of G and is denoted by γtch(G). In this paper, we discuss the chromatic total domination number for standard graphs.

Secure total domination in graphs: Bounds and complexity

A total dominating set of an undirected graph G=(V,E) is a set S⊆V of vertices such that each vertex of G is adjacent to at least one vertex of S. A secure total dominating set of G is a set D⊆V of vertices such that D is a total dominating set of G and each vertex v∈V∖D is adjacent to at least one vertex u∈D with the property that the set (D∖{u})∪{v} is a total dominating set of G. The secure total domination number γst(G) of G is the minimum cardinality of a secure total dominating set of G. We establish bounds on the secure total domination number. In particular, we show that every graph G with no isolated vertices satisfies γst(G)≤2α(G), where α(G) is the independence number of G. Further, we study the problem of finding the secure total domination number. We show that the decision version of the problem is NP-complete for chordal bipartite graphs, planar bipartite graphs with arbitrary large girth and maximum degree 3, split graphs and graphs of separability at most 2. Finally, we show that the optimisation version of the problem can be approximated in polynomial time within a factor of cln|V| for some constant c>1 and cannot be approximated in polynomial time within a factor of c′ln|V| for some constant c′<1, unless P=NP.

Changing and Unchanging in m-Eternal Total Domination in Graphs

The Eternal dominating set of a graph is defined as a set of guards located at vertices, required to protect the vertices of the graph against infinitely long sequences of attacks, such that the configuration of guards induces a dominating set at all times. The eternal m-security number is defined as the minimum number of guards to handle an arbitrary sequence of single attacks using multiple-guard shifts. Kloster-meyer and Mynhardt introduced the concept of an m-eternal total dominating set, which further requires that each vertex with a guard to be adjacent to a vertex with a guard. They defined the m-eternal total domination number of a graph G denoted by γmt∞(G) as the minimum number of guards to handle an arbitrary sequence of single attacks using multiple guard shifts and the configuration of guards always induces a total dominating set. In this paper we examine the effects on γmt∞(G) when G is modified by deleting a vertex which is a non support vertex.

M-Eternal total domination in graphs

Eternal domination of a graph requires the vertices of the graph to be protected, against infinitely long sequences of attacks, by guards located at vertices, with the requirement that the configuration of guards induces a dominating set at all times. Klostermeyer and Mynhardt studied some variations of this concept in which the configuration of guards induces total dominating sets (TDSs). They considered two models of the problem, one in which only one guard moves at a time and one in which more than one guard may move simultaneously. In this paper, we consider the model in which all guards move simultaneously when the configuration of guards induces TDSs and we study this model for some classes of graphs.

Bounds on weak and strong total domination in graphs

A set $D$ of vertices in a graph $G=(V,E)$ is a total dominating set if every vertex of $G$ is adjacent to some vertex in $D$. A total dominating set $D$ of $G$ is said to be weak if every vertex $v\in V-D$ is adjacent to a vertex $u\in D$ such that $d_{G}(v)\geq d_{G}(u)$. The weak total domination number $\gamma_{wt}(G)$ of $G$ is the minimum cardinality of a weak total dominating set of $G$. A total dominating set $D$ of $G$ is said to be strong if every vertex $v\in V-D$ is adjacent to a vertex $u\in D$ such that $d_{G}(v)\leq d_{G}(u)$. The strong total domination number $\gamma_{st}(G)$ of $G$ is the minimum cardinality of a strong total dominating set of $G$. We present some bounds on weak and strong total domination number of a graph.

Weakly connected total domination in graphs

Weakly connected total domination in graphs

A note on neighborhood total domination in graphs

Let G=(V,E) be a graph without isolated vertices. A dominating set S of G is called a neighborhood total dominating set (or just NTDS) if the induced subgraph G[N(S)] has no isolated vertex. The minimum cardinality of a NTDS of G is called the neighborhood total domination number of G and is denoted by γ nt(G). In this paper, we obtain sharp bounds for the neighborhood total domination number of a tree. We also prove that the neighborhood total domination number is equal to the domination number in several classes of graphs including grid graphs.

Bounds on locating total domination number of the Cartesian product of cycles and paths

The problem of placing monitoring devices in a system in such a way that every site in the safeguard system (including the monitors themselves) is adjacent to a monitor site can be modeled by total domination in graphs. Locating-total dominating sets are of interest when the intruder/fault at a vertex precludes its detection in that location. A total dominating set S of a graph G with no isolated vertex is a locating-total dominating set of G if for every pair of distinct vertices u and v in V−S are totally dominated by distinct subsets of the total dominating set. The locating-total domination number of a graph G is the minimum cardinality of a locating-total dominating set of G. In this paper, we study the bounds on locating-total domination numbers of the Cartesian product Cm□Pn of cycles Cm and paths Pn. Exact values for the locating-total domination number of the Cartesian product C3□Pn are found, and it is shown that for the locating-total domination number of the Cartesian product C4□Pn this number is between ⌈3n2⌉ and ⌈3n2⌉+1 with two sharp bounds.

Transversals in 5-uniform hypergraphs and total domination in graphs with minimum degree five

In [9] Thomassé and Yeo pose the following problem: Find the minimum ck for which every k-uniform hypergraph with n vertices and n edges has a transversal of size at most ckn. A direct consequence of this result is that every graph of order n with minimum degree at least k has a total dominating set of cardinality at most ckn. It is known that c2 = 2/3, c3 = 1/2, and c4 = 3/7. Thomassé and Yeo show that 4/11 ≤ c5 and conjecture that c5 = 4/11. In this paper we show that c5 ≤ 17/44. Thus, 16/44 ≤ c5 ≤ 17/44. More generally we prove that every 5-uniform hypergraph on n vertices and m edges has a transversal with no more than (10n+7m)/44 vertices. Consequently, every graph on n vertices with minimum degree at least five has total domination number at most 17n/44.

Trees with large neighborhood total domination number

In this paper, we continue the study of neighborhood total domination in graphs first studied by Arumugam and Sivagnanam (2011). A neighborhood total dominating set, abbreviated NTD-set, in a graph G is a dominating set S in G with the property that the subgraph induced by the open neighborhood of the set S has no isolated vertex. The neighborhood total domination number, denoted by γnt(G), is the minimum cardinality of a NTD-set of G. Every total dominating set is a NTD-set, implying that γ(G)≤γnt(G)≤γt(G), where γ(G) and γt(G) denote the domination and total domination numbers of G, respectively. Arumugam and Sivagnanam posed the problem of characterizing the connected graphs G of order n≥3 achieving the largest possible neighborhood total domination number, namely γnt(G)=⌈n/2⌉. A partial solution to this problem was presented by Henning and Rad (2013) who showed that 5-cycles and subdivided stars are the only such graphs achieving equality in the bound when n is odd. In this paper, we characterize the extremal trees achieving equality in the bound when n is even. As a consequence of this tree characterization, a characterization of the connected graphs achieving equality in the bound when n is even can be obtained noting that every spanning tree of such a graph belongs to our family of extremal trees.

Total Dominator Colorings and Total Domination in Graphs

A total dominator coloring of a graph $$G$$G is a proper coloring of the vertices of $$G$$G in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number $$\chi _d^t(G)$$?dt(G) of $$G$$G is the minimum number of colors among all total dominator coloring of $$G$$G. A total dominating set of $$G$$G is a set $$S$$S of vertices such that every vertex in $$G$$G is adjacent to at least one vertex in $$S$$S. The total domination number $$\gamma _t(G)$$?t(G) of $$G$$G is the minimum cardinality of a total dominating set of $$G$$G. We establish lower and upper bounds on the total dominator chromatic number of a graph in terms of its total domination number. In particular, we show that every graph $$G$$G with no isolated vertex satisfies $$\gamma _t(G) \le \chi _d^t(G) \le \gamma _t(G) + \chi (G)$$?t(G)≤?dt(G)≤?t(G)+?(G), where $$\chi (G)$$?(G) denotes the chromatic number of $$G$$G. We establish properties of total dominator colorings in trees. We characterize the trees $$T$$T for which $$\gamma _t(T) = \chi _d^t(T)$$?t(T)=?dt(T). We prove that if $$T$$T is a tree of $$n \ge 2$$n?2 vertices, then $$\chi _d^t(T) \le 2(n+1)/3$$?dt(T)≤2(n+1)/3 and we characterize the trees achieving equality in this bound.

Lower bounds on the total domination number of a graph

A set $$S$$S of vertices in a graph $$G$$G is a total dominating set if every vertex of $$G$$G is adjacent to a vertex in $$S$$S. The minimum cardinality of a total dominating set of $$G$$G is the total domination number of $$G$$G. Much interest in total domination in graphs has arisen from a computer program Graffiti.pc that has generated several hundred conjectures on total domination (see http://cms.dt.uh.edu/faculty/delavinae/research/wowII). We prove and disprove some conjectures from Graffiti.pc concerning lower bounds on the total domination number of a graph.

Differentiating total domination in graphs: revisited

Differentiating total domination in graphs: revisited

Bounds on neighborhood total domination in graphs

In this paper, we continue the study of neighborhood total domination in graphs first studied by Arumugam and Sivagnanam [S. Arumugam, C. Sivagnanam, Neighborhood total domination in graphs, Opuscula Math. 31 (2011) 519–531]. A neighborhood total dominating set, abbreviated NTD-set, in a graph G is a dominating set S in G with the property that the subgraph induced by the open neighborhood of the set S has no isolated vertex. The neighborhood total domination number, denoted by γnt(G), is the minimum cardinality of a NTD-set of G. Every total dominating set is a NTD-set, implying that γ(G)≤γnt(G)≤γt(G), where γ(G) and γt(G) denote the domination and total domination numbers of G, respectively. We show that if G is a connected graph on n≥3 vertices, then γnt(G)≤(n+1)/2 and we characterize the graphs achieving equality in this bound.