Perfect matchings in paired domination vertex critical graphs

A vertex subset S of a graph G=(V,E) is a paired dominating set if every vertex of G is adjacent to some vertex in S and the subgraph induced by S contains a perfect matching. The paired domination number of G, denoted by γ _pr (G), is the minimum cardinality of a paired dominating set of?G. A?graph with no isolated vertex is called paired domination vertex critical, or briefly γ _pr -critical, if for any vertex v of G that is not adjacent to any vertex of degree one, γ _pr (G?v)<γ _pr (G). A?γ _pr -critical graph G is said to be k-γ _pr -critical if γ _pr (G)=k. In this paper, we firstly show that every 4-γ _pr -critical graph of even order has a perfect matching if it is K _1,5-free and every 4-γ _pr -critical graph of odd order is factor-critical if it is K _1,5-free. Secondly, we show that every 6-γ _pr -critical graph of even order has a perfect matching if it is K _1,4-free.     

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.     

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.     

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).     

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.     

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.     

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.     

Vertices in all minimum paired-dominating sets of?block graphs

Let G=(V,E) be a simple graph without isolated vertices. A set S?V is a paired-dominating set if every vertex in V?S has at least one neighbor in S and the subgraph induced by S contains a perfect matching. In this paper, we present a linear-time algorithm to determine whether a given vertex in a block graph is contained in all its minimum paired-dominating sets.     

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.     

Labelling algorithms for paired-domination problems in block and interval graphs

Let G=(V,E) be a graph without isolated vertices. A set S⊆V is a paired-dominating set if every vertex in V−S is adjacent to a vertex in S and the subgraph induced by S contains a perfect matching. The paired-domination problem is to determine the paired-domination number, which is the minimum cardinality of a paired-dominating set. Motivated by a mistaken algorithm given by Chen, Kang and Ng (Discrete Appl. Math. 155:2077–2086, 2007), we present two linear time algorithms to find a minimum cardinality paired-dominating set in block and interval graphs. In addition, we prove that paired-domination problem is NP-complete for bipartite graphs, chordal graphs, even for split graphs.     

Minimum paired-dominating set in chordal bipartite graphs and perfect elimination bipartite graphs

A set D?V of a graph G=(V,E) is a dominating set of G if every vertex in V?D has at least one neighbor in D. A dominating set D of G is a paired-dominating set of G if the induced subgraph, G[D], has a perfect matching. Given a graph G=(V,E) and a positive integer k, the paired-domination problem is to decide whether G has a paired-dominating set of cardinality at most k. The paired-domination problem is known to be NP-complete for bipartite graphs. In this paper, we, first, strengthen this complexity result by showing that the paired-domination problem is NP-complete for perfect elimination bipartite graphs. We, then, propose a linear time algorithm to compute a minimum paired-dominating set of a chordal bipartite graph, a well studied subclass of bipartite 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.     

Vertices Contained in all or in no Minimum Paired-Dominating Set of a Tree

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. We characterize the set of vertices of a tree that are contained in all, or in no, minimum paired-dominating sets of the tree. Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal.     

Graphs with large paired-domination number

In this paper, we continue the study of paired-domination in graphs introduced by Haynes and Slater (1998) Networks 32: 199–206. A paired-dominating set of a graph G with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of G, denoted by , is the minimum cardinality of a paired-dominating set of G. Let G be a connected graph of order n with minimum degree at least two. Haynes and Slater (1998) Networks 32: 199–206, showed that if n ≥ 6, then . In this paper, we show that there are exactly ten graphs that achieve equality in this bound. For n ≥ 14, we show that and we characterize the (infinite family of) graphs that achieve equality in this bound.Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal.     

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 \(F\cong K_{k+1,k+2}\).     

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.     

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.     

Hardness results and approximation algorithm for total liar’s domination in graphs

In this paper, we initiate the study of total liar’s domination of a graph. A subset L?V of a graph G=(V,E) is called a total liar’s dominating set of G if (i) for all v∈V, |N _G (v)∩L|≥2 and (ii) for every pair u,v∈V of distinct vertices, |(N _G (u)∪N _G (v))∩L|≥3. The total liar’s domination number of a graph G is the cardinality of a minimum total liar’s dominating set of G and is denoted by γ _TLR (G). The Minimum Total Liar’s Domination Problem is to find a total liar’s dominating set of minimum cardinality of the input graph G. Given a graph G and a positive integer k, the Total Liar’s Domination Decision Problem is to check whether G has a total liar’s dominating set of cardinality at most k. In this paper, we give a necessary and sufficient condition for the existence of a total liar’s dominating set in a graph. We show that the Total Liar’s Domination Decision Problem is NP-complete for general graphs and is NP-complete even for split graphs and hence for chordal graphs. We also propose a 2(lnΔ(G)+1)-approximation algorithm for the Minimum Total Liar’s Domination Problem, where Δ(G) is the maximum degree of the input graph G. We show that Minimum Total Liar’s Domination Problem cannot be approximated within a factor of $(\frac{1}{8}-\epsilon)\ln(|V|)$ for any ?>0, unless NP?DTIME(|V|^loglog|V|). Finally, we show that Minimum Total Liar’s Domination Problem is APX-complete for graphs with bounded degree 4.     

Which trees have a differentiating-paired dominating set?

In this paper, we continue the study of paired-domination in graphs introduced by Haynes and Slater (Networks 32 (1998), 199–206). A paired-dominating set of a graph G with no isolated vertex is a dominating set S of vertices whose induced subgraph has a perfect matching. The set S is called a differentiating-paired dominating set if for every pair of distinct vertices u and v in V(G), N[u]∩S≠N[v]∩S, where N[u] denotes the set consisting of u and all vertices adjacent to u. In this paper, we provide a constructive characterization of trees that do not have a differentiating-paired dominating set.     

A characterization of graphs with disjoint dominating and paired-dominating sets

A dominating set of a graph is a set of vertices such that every vertex not in the set is adjacent to a vertex in the set, while a paired-dominating set of a graph is a set of vertices such that every vertex is adjacent to a vertex in the set and the subgraph induced by the set contains a perfect matching. In this paper, we provide a constructive characterization of graphs whose vertex set can be partitioned into a dominating set and a paired-dominating set.