Total Domination Number Γt Research Articles

Relating the total domination number and the annihilation number for quasi-trees and some composite graphs

The total domination number γt(G) of a graph G is the cardinality of a smallest set D⊆V(G) such that each vertex of G has a neighbor in D. The annihilation number a(G) of G is the largest integer k such that there exist k different vertices in G with the degree sum at most m(G). It is conjectured that γt(G)≤a(G)+1 holds for every nontrivial connected graph G. The conjecture has been proved for graphs with minimum degree at least 3, trees, certain tree-like graphs, block graphs, and cactus graphs. In the main result of this paper it is proved that the conjecture holds for quasi-trees. The conjecture is verified also for some graph constructions including bijection graphs, Mycielskians, and the newly introduced universally-identifying graphs.

Domination versus total domination in claw-free cubic graphs

A set S of vertices in a graph G is a dominating set if every vertex not in S is adjacent to a vertex in S. If, in addition, every vertex in S is adjacent to some other vertex in S, then S is a total dominating set. The domination number γ(G) of G is the minimum cardinality of a dominating set in G, while the total domination number γt(G) of G is the minimum cardinality of total dominating set in G. A claw-free graph is a graph that does not contain K1,3 as an induced subgraph. Let G be a connected, claw-free, cubic graph of order n. We show that if we exclude two graphs, then γt(G)γ(G)≤127, and this bound is best possible. In order to prove this result, we prove that if we exclude four graphs, then γt(G)≤37n, and this bound is best possible. These bounds improve previously best known results due to Favaron and Henning (2008) [7], Southey and Henning (2010) [19].

A new upper bound on the total domination number in graphs with minimum degree six

A total dominating set in a graph G is a set of vertices of G such that every vertex is adjacent to a vertex of the set. The total domination number γt(G) is the minimum cardinality of a dominating set in G. Thomassé and Yeo (2007) conjectured that if G is a graph on n vertices with minimum degree at least 5, then γt(G)≤411n. In this paper, it is shown that the Thomassé–Yeo conjecture holds with strict inequality if the minimum degree at least 6. More precisely, it is proven that if G is a graph of order n with δ(G)≥6, then γt(G)≤513814145n<411−12510n. This improves the best known upper bounds to date on the total domination number of a graph with minimum degree at least 6.

Total Domination on Some Graph Operators

Let G=(V,E) be a graph; a set D⊆V is a total dominating set if every vertex v∈V has, at least, one neighbor in D. The total domination number γt(G) is the minimum cardinality among all total dominating sets. Given an arbitrary graph G, we consider some operators on this graph; S(G),R(G), and Q(G), and we give bounds or the exact value of the total domination number of these new graphs using some parameters in the original graph G.

Semitotal Domination: New hardness results and a polynomial-time algorithm for graphs of bounded mim-width

A semitotal dominating set of a graph G with no isolated vertex is a dominating set D of G such that every vertex in D is within distance two of another vertex in D. The minimum size γt2(G) of a semitotal dominating set of G is squeezed between the domination number γ(G) and the total domination number γt(G).Semitotal Dominating Set is the problem of finding, given a graph G, a semitotal dominating set of G of size γt2(G). In this paper, we continue the systematic study on the computational complexity of this problem when restricted to special graph classes. In particular, we show that it is solvable in polynomial time for the class of graphs of bounded mim-width by a reduction to Total Dominating Set and we provide several approximation lower bounds for subclasses of subcubic graphs. Moreover, we obtain complexity dichotomies in monogenic classes for the decision versions of Semitotal Dominating Set and Total Dominating Set.Finally, we show that it is NP-complete to recognise the graphs such that γt2(G)=γt(G) and those such that γ(G)=γt2(G), even if restricted to be planar and with maximum degree at most 4, and we provide forbidden induced subgraph characterisations for the graphs hereditarily satisfying either of these two equalities.

DOMINATION AND TOTAL DOMINATION IN INTUITIONISTIC TRIPLE LAYERED SIMPLE FUZZY GRAPH

In this paper, we discussed domination and total domination in Intuitionistic fuzzy graph. We determined the domination number γ and total domination number γt in Intuitionistic Triple in Intuitionistic Triple Layered Simple fuzzy graph (ITLFG) and also verified the existence of 2-domination in intuitionistic Triple Layered simple fuzzy graph Layered Simple fuzzy graph (ITLFG) and also verified the existence of 2-domination in intuitionistic Triple Layered simple fuzzy graph

Roman and Total Domination

A set S of vertices is a total dominating set of a graph G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set is the total domination number γt(G). A Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u with f (u)=0 is adjacent to at least one vertex v of G for which f (v)=2. The minimum of f (V (G))=∑u ∈ V (G) f (u) over all such functions is called the Roman domination number γR (G). We show that γt(G) ≤ γR (G) with equality if and only if γt(G)=2γ(G), where γ(G) is the domination number of G. Moreover, we characterize the extremal graphs for some graph families.

Game total domination subdivision number of a graph

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 domination numberγt(G) is the minimum cardinality of a total dominating set of G. The game total domination subdivision number of a graph G is defined by the following game. Two players 𝒟 and 𝒜, 𝒟 playing first, alternately mark or subdivide an edge of G which is not yet marked nor subdivided. The game ends when all the edges of G are marked or subdivided and results in a new graph G′. The purpose of 𝒟 is to minimize the total domination number γt(G′) of G′ while 𝒜 tries to maximize it. If both 𝒜 and 𝒟 play according to their optimal strategies, γt(G′) is well defined. We call this number the game total domination subdivision number of G and denote it by γgt(G). In this paper we initiate the study of the game total domination subdivision number of a graph and present some (sharp) bounds for this parameter.

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.

Domination and total domination in cubic graphs of large girth

The domination number γ(G) and the total domination number γt(G) of a graph G without an isolated vertex are among the most well-studied parameters in graph theory. While the inequality γt(G)≤2γ(G) is an almost immediate consequence of the definition, the extremal graphs for this inequality are not well understood. Furthermore, even very strong additional assumptions do not allow us to improve the inequality by much.In the present paper we consider the relation of γ(G) and γt(G) for cubic graphs G of large girth. Clearly, in this case γ(G) is at least n(G)/4 where n(G) is the order of G. If γ(G) is close to n(G)/4, then this forces a certain structure within G. We exploit this structure and prove an upper bound on γt(G), which depends on the value of γ(G). As a consequence, we can considerably improve the inequality γt(G)≤2γ(G) for cubic graphs of large girth.

A new lower bound for the total domination number in graphs proving a Graffiti.pc Conjecture

The total domination number γt(G) of a graph G is the minimum cardinality of a set S of vertices so that every vertex of G is adjacent to a vertex in S. Our main result of this paper is that if G is a connected graph and x1,x2,x3∈V(G), then γt(G)≥14(d(x1,x2)+d(x1,x3)+d(x2,x3)). Furthermore if equality holds in this bound, then the multiset {d(x1,x2),d(x1,x3),d(x2,x3)} is equal to {2,3,3} modulo four. As a consequence of this result, we prove a conjecture on the total domination number. To state this conjecture, let B denote the set of vertices of maximum eccentricity in G and let ecc(B) denote the maximum distance in G of a vertex outside B to a vertex of B. The following conjecture is known as Graffiti.pc Conjecture #233 (http://cms.dt.uh.edu/faculty/delavinae/research/wowII): if G is a connected graph of order at least two, then γt(G)≥2(ecc(B)+1)/3. We prove this conjecture. In fact, as a consequence of our main result stated earlier we prove the following much stronger result: if G is a connected graph of order at least two, then γt(G)≥(3ecc(B)+2)/4. We also prove that if G is a connected graph and x1,x2,x3,x4∈V(G), then γt(G)≥18(d(x1,x2)+d(x1,x3)+d(x1,x4)+d(x2,x3)+d(x2,x4)+d(x3,x4)), and this result is best possible.

Equality in a linear Vizing-like relation that relates the size and total domination number of a graph

Let G be a graph, each component of which has order at least 3, and let G have order n, size m, total domination number γt and maximum degree Δ(G). Let Δ=3 if Δ(G)=2 and Δ=Δ(G) if Δ(G)≥3. It is known [M.A. Henning, A linear Vizing-like relation relating the size and total domination number of a graph, J. Graph Theory 49 (2005) 285–290; E. Shan, L. Kang, M.A. Henning, Erratum to: a linear Vizing-like relation relating the size and total domination number of a graph, J. Graph Theory 54 (2007) 350–353] that m≤Δ(n−γt). In this paper we characterize the extremal graphs G satisfying m=Δ(n−γt).

Relating the annihilation number and the total domination number of a tree

A set S of vertices in a graph G is a total dominating set if every vertex of G is adjacent to some vertex in S. The total domination number γt(G) is the minimum cardinality of a total dominating set in G. The annihilation number a(G) is the largest integer k such that the sum of the first k terms of the non-decreasing degree sequence of G is at most the number of edges in G. In this paper, we investigate relationships between the annihilation number and the total domination number of a graph. Let T be a tree of order n≥2. We show that γt(T)≤a(T)+1, and we characterize the extremal trees achieving equality in this bound.

Matchings and total domination subdivision number in graphs

A set S⊆V of vertices in a graph G=(V,E) without isolated vertices is a total dominating set if every vertex of V is adjacent to some vertex in S. The total domination numberγt(G) is the minimum cardinality of a total dominating set of G. The total domination subdivision numbersdγt(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. In this paper we prove that sdγt(G)≤α′(G)+1 for some classes of graphs where α′(G) is the maximum cardinality of a matching of G.

On 3-[formula omitted]-vertex critical graphs of diameter three

A total dominating set of a graph G=(V,E) with no isolated vertex is a set S⊆V such that every vertex is adjacent to a vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γt(G) of G. Let k≥3 be an integer. A graph G with no isolated vertex is k-γt-vertex critical if γt(G)=k and γt(G−v)<k for every vertex v of G that is not adjacent to a vertex of degree one. In this note we study 3-γt-vertex critical graphs of diameter three. We prove that such graphs have order at least 10, and we characterize all 3-γt-vertex critical graphs of order 10. Moreover, we show that for every even number n≥10, there is a 3-γt-vertex critical graph of order n.

On total domination and support vertices of a tree

The total domination number γt(G) of a simple, undirected graph G is the order of a smallest subset D of the vertices of G such that each vertex of G is adjacent to some vertex in D. In this paper we prove two new upper bounds on the total domination number of a tree related to particular support vertices (vertices adjacent to leaves) of the tree. One of these bounds improves a 2004 result of Chellali and Haynes [1]. In addition, we prove some bounds on the total domination ratio of trees.

A New Bound on the Total Domination Subdivision Number

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 domination number γt (G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number $$sd_{\gamma_{t}}(G)$$is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. In this paper we prove that for every simple connected graph G of order n ≥ 3, $${\rm sd}_{\gamma_{t}}(G)\le 3 +{\rm min}\{d_2(v); v\in V \, {\rm and}\, d(v)\ge 2\}$$.

On the total domination subdivision number in some classes of 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 domination numberγt(G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number\(\mathrm {sd}_{\gamma_{t}}(G)\) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. In this paper we prove that \(\mathrm {sd}_{\gamma_{t}}(G)\leq\gamma_{t}(G)+1\) for some classes of graphs.

Total Domination in Graphs with Given Girth

A set S of vertices in a graph G without isolated vertices is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γt (G) of G. In this paper, we establish an upper bound on the total domination number of a graph with minimum degree at least two in terms of its order and girth. We prove that if G is a graph of order n with minimum degree at least two and girth g, then γt (G) ≤ n/2 + n/g, and this bound is sharp. Our proof is an interplay between graph theory and transversals in hypergraphs.

Total domination in 2‐connected graphs and in graphs with no induced 6‐cycles

AbstractA set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γt(G) of G. It is known [J Graph Theory 35 (2000), 21–45] that if G is a connected graph of order n &gt; 10 with minimum degree at least 2, then γt(G) ≤ 4n/7 and the (infinite family of) graphs of large order that achieve equality in this bound are characterized. In this article, we improve this upper bound of 4n/7 for 2‐connected graphs, as well as for connected graphs with no induced 6‐cycle. We prove that if G is a 2‐connected graph of order n &gt; 18, then γt(G) ≤ 6n/11. Our proof is an interplay between graph theory and transversals in hypergraphs. We also prove that if G is a connected graph of order n &gt; 18 with minimum degree at least 2 and no induced 6‐cycle, then γt(G) ≤ 6n/11. Both bounds are shown to be sharp. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 55–79, 2009