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 number γ _t (G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number sd_gt(G)\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 sd_gt(G) ￡ g_t(G)+1\mathrm {sd}_{\gamma_{t}}(G)\leq\gamma_{t}(G)+1 for some classes of graphs.     

Total restrained domination in claw-free graphs

A set S of vertices in a graph G=(V,E) is a total restrained dominating set (TRDS) of G if every vertex of G is adjacent to a vertex in S and every vertex of V−S is adjacent to a vertex in V−S. The total restrained domination number of G, denoted by γ _tr (G), is the minimum cardinality of a TRDS of G. In this paper we characterize the claw-free graphs G of order n with γ _tr (G)=n. Also, we show that γ _tr (G)≤n−Δ+1 if G is a connected claw-free graph of order n≥4 with maximum degree Δ≤n−2 and minimum degree at least 2 and characterize those graphs which achieve this bound.     

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.     

Paired versus double domination in K 1,r -free graphs

A vertex in G is said to dominate itself and its neighbors. A subset S of vertices is a dominating set if S dominates every vertex of G. A paired-dominating set is a dominating set whose induced subgraph contains a perfect matching. The paired-domination number of a graph G, denoted by γ _pr(G), is the minimum cardinality of a paired-dominating set in G. A subset S?V(G) is a double dominating set of G if S dominates every vertex of G at least twice. The minimum cardinality of a double dominating set of G is the double domination number γ _×2(G). A claw-free graph is a graph that does not contain K _1,3 as an induced subgraph. Chellali and Haynes (Util. Math. 67:161–171, 2005) showed that for every claw-free graph G, we have γ _pr(G)≤γ _×2(G). In this paper we extend this result by showing that for r≥2, if G is a connected graph that does not contain K _1,r as an induced subgraph, then $\gamma_{\mathrm{pr}}(G)\le ( \frac{2r^{2}-6r+6}{r(r-1)} )\gamma_{\times2}(G)$ .     

On domination number of Cartesian product of directed paths

Let γ(G) denote the domination number of a digraph G and let P _m □P _n denote the Cartesian product of P _m and P _n , the directed paths of length m and n. In this paper, we give a lower and upper bound for γ(P _m □P _n ). Furthermore, we obtain a necessary and sufficient condition for P _m □P _n to have efficient dominating set, and determine the exact values: γ(P ₂□P _n )=n, g(P₃\square P_n)=n+é\fracn4ù\gamma(P_{3}\square P_{n})=n+\lceil\frac{n}{4}\rceil, g(P₄\square P_n)=n+é\frac2n3ù\gamma(P_{4}\square P_{n})=n+\lceil\frac{2n}{3}\rceil, γ(P ₅□P _n )=2n+1 and g(P₆\square P_n)=2n+é\fracn+23ù\gamma(P_{6}\square P_{n})=2n+\lceil\frac{n+2}{3}\rceil.     

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.     

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.     

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.     

Some results on the injective chromatic number of?graphs

A k-coloring of a graph G=(V,E) is a mapping c:V??{1,2,??,k}. The coloring c is injective if, for every vertex v??V, all the neighbors of v are assigned with distinct colors. The injective chromatic number ?? _i (G) of G is the smallest k such that G has an injective k-coloring. In this paper, we prove that every K ₄-minor free graph G with maximum degree ????1 has $\chi_{i}(G)\le \lceil \frac{3}{2}\Delta\rceil$ . Moreover, some related results and open problems are given.     

Group testing in graphs

This paper studies the group testing problem in graphs as follows. Given a graph G=(V,E), determine the minimum number t(G) such that t(G) tests are sufficient to identify an unknown edge e with each test specifies a subset X⊆V and answers whether the unknown edge e is in G[X] or not. Damaschke proved that ⌈log ₂ e(G)⌉≤t(G)≤⌈log ₂ e(G)⌉+1 for any graph G, where e(G) is the number of edges of G. While there are infinitely many complete graphs that attain the upper bound, it was conjectured by Chang and Hwang that the lower bound is attained by all bipartite graphs. Later, they proved that the conjecture is true for complete bipartite graphs. Chang and Juan verified the conjecture for bipartite graphs G with e(G)≤2⁴ or for k≥5. This paper proves the conjecture for bipartite graphs G with e(G)≤2⁵ or for k≥6. Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday. J.S.-t.J. is supported in part by the National Science Council under grant NSC89-2218-E-260-013. G.J.C. is supported in part by the National Science Council under grant NSC93-2213-E002-28. Taida Institute for Mathematical Sciences, National Taiwan University, Taipei 10617, Taiwan. National Center for Theoretical Sciences, Taipei Office.     

On the distance paired domination of generalized Petersen graphs <Emphasis Type="Italic">P</Emphasis>(<Emphasis Type="Italic">n</Emphasis>,1) and <Emphasis Type="Italic">P</Emphasis>(<Emphasis Type="Italic">n</Emphasis>,2)

Let G=(V,E) be a graph without an isolated vertex. A set D⊆V(G) is a k -distance paired dominating set of G if D is a k-distance dominating set of G and the induced subgraph 〈D〉 has a perfect matching. The minimum cardinality of a k-distance paired dominating set for graph G is the k -distance paired domination number, denoted by γ _p ^k (G). In this paper, we determine the exact k-distance paired domination number of generalized Petersen graphs P(n,1) and P(n,2) for all k≥1.     

Paired-domination number of claw-free odd-regular graphs

A paired-dominating set of a graph G is a dominating set of vertices whose induced subgraph has a perfect matching, while the paired-domination number is the minimum cardinality of a paired-dominating set in the graph, denoted by \(\gamma _{pr}(G)\). Let G be a connected \(\{K_{1,3}, K_{4}-e\}\)-free cubic graph of order n. We show that \(\gamma _{pr}(G)\le \frac{10n+6}{27}\) if G is \(C_{4}\)-free and that \(\gamma _{pr}(G)\le \frac{n}{3}+\frac{n+6}{9(\lceil \frac{3}{4}(g_o+1)\rceil +1)}\) if G is \(\{C_{4}, C_{6}, C_{10}, \ldots , C_{2g_o}\}\)-free for an odd integer \(g_o\ge 3\); the extremal graphs are characterized; we also show that if G is a 2 -connected, \(\gamma _{pr}(G) = \frac{n}{3} \). Furthermore, if G is a connected \((2k+1)\)-regular \(\{K_{1,3}, K_4-e\}\)-free graph of order n, then \(\gamma _{pr}(G)\le \frac{n}{k+1} \), with equality if and only if \(G=L(F)\), where \(F\cong K_{1, 2k+2}\), or k is even and \(F\cong K_{k+1,k+2}\).     

The total {k}-domatic number of a graph

For a positive integer k, a total {k}-dominating function of a graph G is a function f from the vertex set V(G) to the set {0,1,2,…,k} such that for any vertex v∈V(G), the condition ∑ _u∈N(v) f(u)≥k is fulfilled, where N(v) is the open neighborhood of v. A set {f ₁,f ₂,…,f _d } of total {k}-dominating functions on G with the property that ?_i=1^df_i(v) ￡ k\sum_{i=1}^{d}f_{i}(v)\le k for each v∈V(G), is called a total {k}-dominating family (of functions) on G. The maximum number of functions in a total {k}-dominating family on G is the total {k}-domatic number of G, denoted by d_t^{k}(G)d_{t}^{\{k\}}(G). Note that d_t^{1}(G)d_{t}^{\{1\}}(G) is the classic total domatic number d _t (G). In this paper we initiate the study of the total {k}-domatic number in graphs and we present some bounds for d_t^{k}(G)d_{t}^{\{k\}}(G). Many of the known bounds of d _t (G) are immediate consequences of our results.     

Upper paired-domination in claw-free graphs

A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The maximum cardinality of a minimal paired-dominating set of G is the upper paired-domination number of G, denoted by Γ_pr(G). We establish bounds on Γ_pr(G) for connected claw-free graphs G in terms of the number n of vertices in G with given minimum degree δ. We show that Γ_pr(G)≤4n/5 if δ=1 and n≥3, Γ_pr(G)≤3n/4 if δ=2 and n≥6, and Γ_pr(G)≤2n/3 if δ≥3. All these bounds are sharp. Further, if n≥6 the graphs G achieving the bound Γ_pr(G)=4n/5 are characterized, while for n≥9 the graphs G with δ=2 achieving the bound Γ_pr(G)=3n/4 are characterized.     

A 6.55 factor primal-dual approximation algorithm for the connected facility location problem

In the connected facility location (ConFL) problem, we are given a graph G=(V,E) with nonnegative edge cost c _e on the edges, a set of facilities ??V, a set of demands (i.e., clients) $\mathcal{D}\subseteq VIn the connected facility location (ConFL) problem, we are given a graph G=(V,E) with nonnegative edge cost c _e on the edges, a set of facilities ℱ⊆V, a set of demands (i.e., clients) D í V\mathcal{D}\subseteq V , and a parameter M≥1. Each facility i has a nonnegative opening cost f _i and each client j has d _j units of demand. Our objective is to open some facilities, say F⊆ℱ, assign each demand j to some open facility i(j)∈F and connect all open facilities using a Steiner tree T such that the total cost, which is ?_{i ? F}f_i+?_{j ? D}d_jc_i(j)j+M?_{e ? T}c_e\sum_{i\in F}f_{i}+\sum_{j\in \mathcal{D}}d_{j}c_{i(j)j}+M\sum_{e\in T}c_{e} , is minimized. We present a primal-dual 6.55-approximation algorithm for the ConFL problem which improves the previous primal-dual 8.55-approximation algorithm given by Swamy and Kumar (Algorithmica 40:245–269, 2004).     

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.     

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.     

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).     

A lower bound of the surviving rate of a planar graph with girth at least seven

Let G be a connected graph with n≥2 vertices. Suppose that a fire breaks out at a vertex v of G. A firefighter starts to protect vertices. At each time interval, the firefighter protects one vertex not yet on fire. At the end of each time interval, the fire spreads to all the unprotected vertices that have a neighbor on fire. Let sn(v) denote the maximum number of vertices in G that the firefighter can save when a fire breaks out at vertex v. The surviving rate ρ(G) of G is defined to be ∑ _v∈V(G)sn(v)/n ², which is the average proportion of saved vertices. In this paper, we show that if G is a planar graph with n≥2 vertices and having girth at least 7, then $\rho(G)>\frac{1}{301}$ .     

Paired-domination in generalized claw-free graphs

In this paper, we continue the study of paired-domination in graphs introduced by Haynes and Slater (Networks 32 (1998) 199–206). A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired-domination number of G, denoted by , is the minimum cardinality of a paired-dominating set of G. If G does not contain a graph F as an induced subgraph, then G is said to be F-free. Haynes and Slater (Networks 32 (1998) 199–206) showed that if G is a connected graph of order , then and this bound is sharp for graphs of arbitrarily large order. Every graph is -free for some integer a ≥ 0. We show that for every integer a ≥ 0, if G is a connected -free graph of order n ≥ 2, then with infinitely many extremal graphs.