On the upper total domination number of Cartesian products of graphs

In this paper we continue the investigation of total domination in Cartesian products of graphs first studied in (Henning, M.A., Rall, D.F. in Graphs Comb. 21:63–69, 2005). A set S of vertices in a graph G is a total dominating set of G if every vertex in G is adjacent to some vertex in S. The maximum cardinality of a minimal total dominating set of G is the upper total domination number of G, denoted by Γ _t (G). We prove that the product of the upper total domination numbers of any graphs G and H without isolated vertices is at most twice the upper total domination number of their Cartesian product; that is, Γ _t (G)Γ _t (H)≤2Γ _t (G □ H). Research of M.A. Henning supported in part by the South African National Research Foundation and the University of KwaZulu-Natal.     

The total domination subdivision number in graphs with no induced 3-cycle and 5-cycle

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 γ _t (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. Favaron, Karami, Khoeilar and Sheikholeslami (J. Comb. Optim. 20:76–84, 2010a) conjectured that: For any connected graph G of order n≥3, $\mathrm{sd}_{\gamma_{t}}(G)\le \gamma_{t}(G)+1$ . In this paper we use matching to prove this conjecture for graphs with no 3-cycle and 5-cycle. In particular this proves the conjecture for bipartite graphs.     

On the total outer-connected domination in graphs

A set S of vertices of a graph G is a total outer-connected dominating set if every vertex in V(G) is adjacent to some vertex in S and the subgraph induced by V?S is connected. The total outer-connected domination number γ _toc (G) is the minimum size of such a set. We give some properties and bounds for γ _toc in general graphs and in trees. For graphs of order n, diameter 2 and minimum degree at least 3, we show that $\gamma_{toc}(G)\le \frac{2n-2}{3}$ and we determine the extremal graphs.     

Disjunctive total domination in graphs

Let \(G\) be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, \(\gamma _t(G)\). A set \(S\) of vertices in \(G\) is a disjunctive total dominating set of \(G\) if every vertex is adjacent to a vertex of \(S\) or has at least two vertices in \(S\) at distance \(2\) from it. The disjunctive total domination number, \(\gamma ^d_t(G)\), is the minimum cardinality of such a set. We observe that \(\gamma ^d_t(G) \le \gamma _t(G)\). We prove that if \(G\) is a connected graph of order \(n \ge 8\), then \(\gamma ^d_t(G) \le 2(n-1)/3\) and we characterize the extremal graphs. It is known that if \(G\) is a connected claw-free graph of order \(n\), then \(\gamma _t(G) \le 2n/3\) and this upper bound is tight for arbitrarily large \(n\). We show this upper bound can be improved significantly for the disjunctive total domination number. We show that if \(G\) is a connected claw-free graph of order \(n > 14\), then \(\gamma ^d_t(G) \le 4n/7\) and we characterize the graphs achieving equality in this bound.     

On the power domination number of the generalized Petersen graphs

The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well-known domination problem in graphs. Following a set of rules for power system monitoring, a set S of vertices is defined to be a power dominating set of a graph if every vertex and every edge in the system is monitored by the set S. The minimum cardinality of a power dominating set of G is the power domination number γ _p (G). In this paper, we investigate the power domination number for the generalized Petersen graphs, presenting both upper bounds for such graphs and exact results for a subfamily of generalized Petersen graphs.     

Paired domination versus domination and packing number in graphs

Journal of Combinatorial Optimization - Given a graph $$G=(V(G), E(G))$$ , the size of a minimum dominating set, minimum paired dominating set, and a minimum total dominating set of a graph G are...     

Signed total Roman domination in graphs

Let \(G\) be a finite and simple graph with vertex set \(V(G)\). A signed total Roman dominating function (STRDF) on a graph \(G\) is a function \(f:V(G)\rightarrow \{-1,1,2\}\) satisfying the conditions that (i) \(\sum _{x\in N(v)}f(x)\ge 1\) for each vertex \(v\in V(G)\), where \(N(v)\) is the neighborhood of \(v\), and (ii) every vertex \(u\) for which \(f(u)=-1\) is adjacent to at least one vertex \(v\) for which \(f(v)=2\). The weight of an SRTDF \(f\) is \(\sum _{v\in V(G)}f(v)\). The signed total Roman domination number \(\gamma _{stR}(G)\) of \(G\) is the minimum weight of an STRDF on \(G\). In this paper we initiate the study of the signed total Roman domination number of graphs, and we present different bounds on \(\gamma _{stR}(G)\). In addition, we determine the signed total Roman domination number of some classes of graphs.     

On the outer-connected domination in graphs

A set S of vertices of a graph G is an outer-connected dominating set if every vertex not in S is adjacent to some vertex in S and the subgraph induced by V?S is connected. The outer-connected domination number $\widetilde{\gamma}_{c}(G)$ is the minimum size of such a set. We prove that if δ(G)≥2 and diam?(G)≤2, then $\widetilde{\gamma}_{c}(G)\le (n+1)/2$ , and we study the behavior of $\widetilde{\gamma}_{c}(G)$ under an edge addition.     

Edge lifting and total domination in graphs

Let u and v be vertices of a graph G, such that the distance between u and v is two and x is a common neighbor of u and v. We define the edge lift of uv off x as the process of removing edges ux and vx while adding the edge uv to G. In this paper, we investigate the effect that edge lifting has on the total domination number of a graph. Among other results, we show that there are no trees for which every possible edge lift decreases the total domination number and that there are no trees for which every possible edge lift leaves the total domination number unchanged. Trees for which every possible edge lift increases the total domination number are characterized.     

k-tuple total domination in cross products of graphs

For k??1 an integer, a set S of vertices in a graph G with minimum degree at least?k is a k-tuple total dominating set of G if every vertex of G is adjacent to at least k vertices in S. The minimum cardinality of a k-tuple total dominating set of G is the k-tuple total domination number of G. When k=1, the k-tuple total domination number is the well-studied total domination number. In this paper, we establish upper and lower bounds on the k-tuple total domination number of the cross product graph G×H for any two graphs G and H with minimum degree at least?k. In particular, we determine the exact value of the k-tuple total domination number of the cross product of two complete graphs.     

On total and edge coloring some Kneser graphs

Journal of Combinatorial Optimization - In this work, we investigate the total and edge colorings of the Kneser graphs K(n, s). We prove that the sparse case of Kneser graphs, the odd...     

An efficient algorithm for distance total domination in block graphs

The \(k\)-distance total domination problem is to find a minimum vertex set \(D\) of a graph such that every vertex of the graph is within distance \(k\) from some vertex of \(D\) other than itself, where \(k\) is a fixed positive integer. In the present paper, by using a labeling method, we design an efficient algorithm for solving the \(k\)-distance total domination problem on block graphs, a superclass of trees.     

On the total {k}-domination number of Cartesian products of graphs

Let γ _t ^{k}(G) denote the total {k}-domination number of graph G, and let denote the Cartesian product of graphs G and H. In this paper, we show that for any graphs G and H without isolated vertices, . As a corollary of this result, we have for all graphs G and H without isolated vertices, which is given by Pak Tung Ho (Util. Math., 2008, to appear) and first appeared as a conjecture proposed by Henning and Rall (Graph. Comb. 21:63–69, 2005). The work was supported by NNSF of China (No. 10701068 and No. 10671191).     

Restricted domination parameters in graphs

In a graph G, a vertex dominates itself and its neighbors. A subset S ⊂eqV(G) is an m-tuple dominating set if S dominates every vertex of G at least m times, and an m-dominating set if S dominates every vertex of G−S at least m times. The minimum cardinality of a dominating set is γ, of an m-dominating set is γ _m , and of an m-tuple dominating set is mtupledom. For a property π of subsets of V(G), with associated parameter f_π, the k-restricted π-number r _k (G,f_π) is the smallest integer r such that given any subset K of (at most) k vertices of G, there exists a π set containing K of (at most) cardinality r. We show that for 1< k < n where n is the order of G: (a) if G has minimum degree m, then r _k (G,γ _m ) < (mn+k)/(m+1); (b) if G has minimum degree 3, then r _k (G,γ) < (3n+5k)/8; and (c) if G is connected with minimum degree at least 2, then r _k (G,ddom) < 3n/4 + 2k/7. These bounds are sharp. Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal.     

Signed Roman domination in graphs

In this paper we continue the study of Roman dominating functions in graphs. A signed Roman dominating function (SRDF) on a graph G=(V,E) is a function f:V→{?1,1,2} satisfying the conditions that (i) the sum of its function values over any closed neighborhood is at least one and (ii) for every vertex u for which f(u)=?1 is adjacent to at least one vertex v for which f(v)=2. The weight of a SRDF is the sum of its function values over all vertices. The signed Roman domination number of G is the minimum weight of a SRDF in G. We present various lower and upper bounds on the signed Roman domination number of a graph. Let G be a graph of order n and size m with no isolated vertex. We show that $\gamma _{\mathrm{sR}}(G) \ge\frac{3}{\sqrt{2}} \sqrt{n} - n$ and that γ _sR(G)≥(3n?4m)/2. In both cases, we characterize the graphs achieving equality in these bounds. If G is a bipartite graph of order n, then we show that $\gamma_{\mathrm{sR}}(G) \ge3\sqrt{n+1} - n - 3$ , and we characterize the extremal graphs.     

An upper bound on the total restrained domination number of a tree

Let G=(V,E) be a graph. A set of vertices S?V is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of $V-\nobreak S$ is adjacent to a vertex in V?S. The total restrained domination number of G, denoted by γ _tr (G), is the smallest cardinality of a total restrained dominating set of G. A support vertex of a graph is a vertex of degree at least two which is adjacent to a leaf. We show that $\gamma_{\mathit{tr}}(T)\leq\lfloor\frac{n+2s+\ell-1}{2}\rfloor$ where T is a tree of order n≥3, and s and ? are, respectively, the number of support vertices and leaves of T. We also constructively characterize the trees attaining the aforementioned bound.     

Restrained domination in cubic graphs

Let G=(V,E) be a graph. A set S⊆V is a restrained dominating set if every vertex in V−S is adjacent to a vertex in S and to a vertex in V−S. The restrained domination number of G, denoted γ _r (G), is the smallest cardinality of a restrained dominating set of G. A graph G is said to be cubic if every vertex has degree three. In this paper, we study restrained domination in cubic graphs. We show that if G is a cubic graph of order n, then g_r(G) 3 \fracn4\gamma_{r}(G)\geq \frac{n}{4} , and characterize the extremal graphs achieving this lower bound. Furthermore, we show that if G is a cubic graph of order n, then g_r(G) ￡ \frac5n11.\gamma _{r}(G)\leq \frac{5n}{11}. Lastly, we show that if G is a claw-free cubic graph, then γ _r (G)=γ(G).     

Generalized perfect domination in graphs

Let k be a positive integer and G=(V,E) be a graph. A vertex subset D of a graph G is called a perfect k-dominating set of G, if every vertex v of G, not in D, is adjacent to exactly k vertices of D. The minimum cardinality of a perfect k-dominating set of G is the perfect k-domination number γ _kp (G). In this paper, we give characterizations of graphs for which γ _kp (G)=γ(G)+k?2 and prove that the perfect k-domination problem is NP-complete even when restricted to bipartite graphs and chordal graphs. Also, by using dynamic programming techniques, we obtain an algorithm to determine the perfect k-domination number of trees.