Total Domination Number Research Articles

Total domination in planar graphs of diameter two

MacGillivary and Seyffarth [G. MacGillivray, K. Seyffarth, Domination numbers of planar graphs, J. Graph Theory 22 (1996) 213–229] proved that planar graphs of diameter two have domination number at most three. Goddard and Henning [W. Goddard, M.A. Henning, Domination in planar graphs with small diameter, J. Graph Theory 40 (2002) 1–25] showed that there is a unique planar graph of diameter two with domination number three. It follows that the total domination number of a planar graph of diameter two is at most three. In this paper, we consider the problem of characterizing planar graphs with diameter two and total domination number three. We say that a graph satisfies the domination-cycle property if there is some minimum dominating set of the graph not contained in any induced 5-cycle. We characterize the planar graphs with diameter two and total domination number three that satisfy the domination-cycle property and show that there are exactly thirty-four such planar graphs.

Bounding the total domination subdivision number of a graph in terms of its order

The total domination subdivision number $\mathrm{sd}_{\gamma _{t}}(G)$ of a graph 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 $\mathrm{sd}_{\gamma_{t}}(G)\leq \lfloor\frac{2n}{3}\rfloor$ for any simple connected graph G of order n?3 other than K 4. We also determine all simple connected graphs G with $\mathrm{sd}_{\gamma_{t}}(G)=\lfloor\frac{2n}{3}\rfloor$ .

Domination in a digraph and in its reverse

Let D be a digraph. By γ ( D ) we denote the domintaion number of D and by D − we denote a digraph obtained by reversing all the arcs of D . In this paper we prove that for every δ ≥ 3 and k ≥ 1 there exists a simple strongly connected δ -regular digraph D δ , k such that γ ( D δ , k − ) − γ ( D δ , k ) = k . Analogous theorem is obtained for total domination number provided that δ ≥ 4 .

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, $${\rm sd}_{\gamma_{t}}(G)\le 3 +{\rm min}\{d_2(v); v\in V \, {\rm and}\, d(v)\ge 2\}$$.

Signed (total) domination numbers and Laplacian spectrum of graphs

Let G = (V, E) be a graph of order n and size m. The open neighborhood N(v) of a vertex v ∈ V is the set of all vertices adjacent to v. Its closed neighborhood is N[v] = N(v) ∪ {v}. A signed (total) dominating function on G, is a function f : V → {−1, 1} such that ∑ x∈N[v] f (x) ≥ 1 (∑ x∈N(v) f (x) ≥ 1) for each v ∈ V. The weight of a signed (total) dominating function f is ω(f) = ∑ v∈V f (v). The minimum weight taken over all signed (total) dominating function f of G, denoted by γ s (G)(γ st (G)), is called the signed (total) domination number of G. In the paper, we give bounds on the Laplacian spectrum of G involving the signed domination and signed total domination numbers.

On total domination vertex critical graphs of high connectivity

A graph is called γ -critical if the removal of any vertex from the graph decreases the domination number, while a graph with no isolated vertex is γ t -critical if the removal of any vertex that is not adjacent to a vertex of degree 1 decreases the total domination number. A γ t -critical graph that has total domination number k , is called k - γ t -critical. In this paper, we introduce a class of k - γ t -critical graphs of high connectivity for each integer k ≥ 3 . In particular, we provide a partial answer to the question “Which graphs are γ -critical and γ t -critical or one but not the other?” posed in a recent work [W. Goddard, T.W. Haynes, M.A. Henning, L.C. van der Merwe, The diameter of total domination vertex critical graphs, Discrete Math. 286 (2004) 255–261].

Total Domination number of Generalized Petersen Graphs

Generalized Petersen graphs are an important class of commonly used interconnection networks and have been studied . The total domination number of generalized Petersen graphs P(m,2) is obtained in this paper.

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.

Lower bounds on signed edge total domination numbers in graphs

The open neighborhood NG(e) of an edge e in a graph G is the set consisting of all edges having a common end-vertex with e. Let f be a function on E(G), the edge set of G, into the set {−1, 1}. If \( \sum\limits_{x \in N_G (e)} {f(x) \geqslant 1} \) for each e ∈ E(G), then f is called a signed edge total dominating function of G. The minimum of the values \( \sum\limits_{e \in E(G)} {f(e)} \), taken over all signed edge total dominating function f of G, is called the signed edge total domination number of G and is denoted by γst′(G). Obviously, γst′(G) is defined only for graphs G which have no connected components isomorphic to K2. In this paper we present some lower bounds for γst′(G). In particular, we prove that γst′(T) ⩾ 2 − m/3 for every tree T of size m ⩾ 2. We also classify all trees T with γst′(T).

Hypergraphs with large transversal number and with edge sizes at least 3

AbstractIt is shown in several manuscripts [Discrete Math 86 (1990), 117–126; Combinatorica 12 (1992), 19–26] that every hypergraph of order n and size m with edge sizes at least 3 has a transversal of size at most (n + m)/4. In this article, we characterize the connected such hypergraphs that achieve equality in this bound. As a direct consequence of this bound, the total domination of a graph with minimum degree at least 3 is at most one‐half its order. An elegant graph theoretic proof of this result is presented in Archdeacon et al. [J Graph Theory 46 (2004), 207–210]. Using our hypergraph result, we characterize the connected graphs with minimum degree at least 3 and with total domination number exactly one‐half their order. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 326–348, 2008

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 > 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 > 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 > 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

A note on dominating sets and average distance

We show that the total domination number of a simple connected graph is greater than the average distance of the graph minus one-half, and that this inequality is best possible. In addition, we show that the domination number of the graph is greater than two-thirds of the average distance minus one-third, and that this inequality is best possible. Although the latter inequality is a corollary to a result of P. Dankelmann, we present a short and direct proof.

Upper bounds on the upper signed total domination number of graphs

Let G = ( V , E ) be a graph. A function f : V → { − 1 , + 1 } defined on the vertices of G is a signed total dominating function if the sum of its function values over any open neighborhood is at least one. A signed total dominating function f is minimal if there does not exist a signed total dominating function g , f ≠ g , for which g ( v ) ≤ f ( v ) for every v ∈ V . The weight of a signed total dominating function is the sum of its function values over all vertices of G . The upper signed total domination number of G is the maximum weight of a minimal signed total dominating function on G . In this paper we present a sharp upper bound on the upper signed total domination number of an arbitrary graph. This result generalizes previous results for regular graphs and nearly regular graphs.

A constructive characterization of total domination vertex critical graphs

Let G be a graph of order n and maximum degree Δ ( G ) and let γ t ( G ) denote the minimum cardinality of a total dominating set of a graph G . A graph G with no isolated vertex is the total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G − v is less than the total domination number of G . We call these graphs γ t -critical. For any γ t -critical graph G , it can be shown that n ≤ Δ ( G ) ( γ t ( G ) − 1 ) + 1 . In this paper, we prove that: Let G be a connected γ t -critical graph of order n ( n ≥ 3 ), then n = Δ ( G ) ( γ t ( G ) − 1 ) + 1 if and only if G is regular and, for each v ∈ V ( G ) , there is an A ⊆ V ( G ) − { v } such that N ( v ) ∩ A = 0̸ , the subgraph induced by A is 1-regular, and every vertex in V ( G ) − A − { v } has exactly one neighbor in A .

Domination and total domination in complementary prisms

Let G be a graph and ${\overline {G}}$ be the complement of G. The complementary prism $G{\overline {G}}$ of G is the graph formed from the disjoint union of G and ${\overline {G}}$ by adding the edges of a perfect matching between the corresponding vertices of G and ${\overline {G}}$ . For example, if G is a 5-cycle, then $G{\overline {G}}$ is the Petersen graph. In this paper we consider domination and total domination numbers of complementary prisms. For any graph G, $\max\{\gamma(G),\gamma({\overline {G}})\}\le \gamma(G{\overline {G}})$ and $\max\{\gamma_{t}(G),\gamma_{t}({\overline {G}})\}\le \gamma_{t}(G{\overline {G}})$ , where γ(G) and γ t (G) denote the domination and total domination numbers of G, respectively. Among other results, we characterize the graphs G attaining these lower bounds.

Trees with equal total and total restrained domination numbers

For a graph G = (V; E), a set S V (G) is a total dominating set if it is dominating and both hSi has no isolated vertices. The cardinality of a minimum total dominating set in G is the total domination number. A set S V (G) is a total restrained dominating set if it is total dominating and hV (G) Si has no isolated vertices. The cardinality of a minimum total restrained dominating set in G is the total restrained domination number. We characterize all trees for which total domination and total restrained domination numbers are the same.

Secure domination and secure total domination in graphs

A secure (total) dominating set of a graph G = (V; E) is a (total) dominating set X V with the property that for each u 2 V X, there exists x 2 X adjacent to u such that (X fxg) [ fug is (total) dominating. The smallest cardinality of a secure (total) dominating set is the secure (total) domination number s(G) ( st(G)). We characterize graphs with equal total and secure total domination numbers. We show that if G has minimum degree at least two, then st(G) s(G). We also show that st(G) is at most twice the clique covering number of G, and less than three times the independence number. With the exception of the independence number bound, these bounds are sharp.

Total domination in K₅- and K₆-covered graphs

Graphs and Algorithms A graph G is Kr-covered if each vertex of G is contained in a Kr-clique. Let $\gamma_t(G)$ denote the total domination number of G. It has been conjectured that every Kr-covered graph of order n with no Kr-component satisfies $\gamma_t(G) \le \frac{2n}{r+1}$. We prove that this conjecture is true for r = 5 and 6.

A Transition from Total Domination in Graphs to Transversals in Hypergraphs

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 a vertex in S. The total domination number of G is the minimum cardinality of a total dominating set in G. The transversal number of a hypergraph is the minimum number of vertices meeting every edge. We observe that total domination in graphs can be translated to the problem of finding transversals in hypergraphs. In this paper we survey bounds on the total domination of a graph in terms of the order of the graph, and provide a transition from total domination in graphs to transversals in hypergraphs.