A linear-time algorithm for weighted paired-domination on block graphs

Journal of Combinatorial Optimization - In a graph $$G = (V,E)$$ , a set $$S\subseteq V(G)$$ is said to be a dominating set of G if every vertex not in S is adjacent to a vertex in S. Let G[S]...     

Algorithmic aspect on the minimum (weighted) doubly resolving set problem of graphs

Journal of Combinatorial Optimization - Let G be a simple graph, where each vertex has a nonnegative weight. A vertex subset S of G is a doubly resolving set (DRS) of G if for every pair of...     

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.     

An inequality that relates the size of a bipartite graph with its order and restrained domination number

Let \(G=(V,E)\) be a graph. A set \(S\subseteq V\) 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 \(\gamma _{r}(G)\), is the smallest cardinality of a restrained dominating set of \(G\). Consider a bipartite graph \(G\) of order \(n\ge 4,\) and let \(k\in \{2,3,...,n-2\}.\) In this paper we will show that if \(\gamma _{r}(G)=k\), then \(m\le ((n-k)(n-k+6)+4k-8)/4\). We will also show that this bound is best possible.     

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.     

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.     

Metric dimension of some distance-regular graphs

A resolving set of a graph is a set of vertices with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. In this paper, we construct a resolving set of Johnson graphs, doubled Odd graphs, doubled Grassmann graphs and twisted Grassmann graphs, respectively, and obtain the upper bounds on the metric dimension of these graphs.     

A better constant-factor approximation for weighted dominating set in unit disk graph

This paper presents a (10+ε)-approximation algorithm to compute minimum-weight connected dominating set (MWCDS) in unit disk graph. MWCDS is to select a vertex subset with minimum weight for a given unit disk graph, such that each vertex of the graph is contained in this subset or has a neighbor in this subset. Besides, the subgraph induced by this vertex subset is connected. Our algorithm is composed of two phases: the first phase computes a dominating set, which has approximation ratio 6+ε (ε is an arbitrary positive number), while the second phase connects the dominating sets computed in the first phase, which has approximation ratio 4. This work is supported in part by National Science Foundation under grant CCF-9208913 and CCF-0728851; and also supported by NSFC (60603003) and XJEDU.     

Local Optimality and Its Application on Independent Sets for k-claw Free Graphs

Given a graph G = (V,E), we define the locally optimal independent sets asfollows. Let S be an independent set and T be a subset of V such that S T = and (S) T, where (S) is defined as the neighbor set of S. A minimum dominating set of S in T is defined as T_D(S) T such that every vertex of S is adjacent to a vertex inTD(S) and TD(S) has minimum cardinality. An independent setI is called r-locally optimal if it is maximal and there exists noindependent set S V\I with |ID (S)| r such that|S| >|I (S)|.In this paper, we demonstrate that for k-claw free graphs ther-locally optimal independent sets is found in polynomial timeand the worst case is bounded by , where I and I* are a locally optimal and an optimal independent set,respectively. This improves the best published bound by Hochbaum (1983) bynearly a factor of two. The bound is proved by LP duality and complementaryslackness. We provide an efficientO(|V|^r+3) algorithm to find an independent set which is notnecessarily r-locally optimal but is guarantteed with the above bound. Wealso present an algorithm to find a r-locally optimal independent set inO(|V|^r(k-1)+3) time.     

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.     

Results on vertex-edge and independent vertex-edge domination

Journal of Combinatorial Optimization - Given a graph $$G = (V,E)$$ , a vertex $$u \in V$$ ve-dominates all edges incident to any vertex of $$N_G[u]$$ . A set $$S \subseteq V$$ is a ve-dominating...     

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.     

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.     

Equality in a bound that relates the size and the restrained domination number of a graph

Let \(G=(V,E)\) be a graph. A set \(S\subseteq V\) 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 \(\gamma _{r}(G)\), is the smallest cardinality of a restrained dominating set of \(G\). The best possible upper bound \(q(n,k)\) is established in Joubert (Discrete Appl Math 161:829–837, 2013) on the size \(m(G)\) of a graph \(G\) with a given order \(n \ge 5\) and restrained domination number \(k \in \{3, \ldots , n-2\}\). We extend this result to include the cases \(k=1,2,n\), and characterize graphs \(G\) of order \(n \ge 1\) and restrained domination number \(k \in \{1,\dots , n-2,n\}\) for which \(m(G)=q(n,k)\).     

Bipartite communities via spectral partitioning

Journal of Combinatorial Optimization - In this paper we are interested in finding communities with bipartite structure. A bipartite community is a pair of disjoint vertex sets S, $$S'$$ such...     

Tagged Probe Interval Graphs

A generalization of interval graph is introduced for cosmid contig mapping of DNA. A graph is a tagged probe interval graph if its vertex set can be partitioned into two subsets of probes and nonprobes, and a closed interval can be assigned to each vertex such that two vertices are adjacent if and only if at least one of them is a probe and one end of its corresponding interval is contained in the interval corresponding to the other vertex. We show that tagged probe interval graphs are weakly triangulated graphs, hence are perfect graphs. For a tagged probe interval graph with a given partition, we give a chordal completion that is consistent to any interval completions with respect to the same vertex partition.     

Steiner tree in k-star caterpillar convex bipartite graphs: a dichotomy

Journal of Combinatorial Optimization - A bipartite graph G(X, Y) whose vertex set is partitioned into X and Y is a convex bipartite graph, if there is an ordering of $$X=(x_1,\ldots...     

Total and paired domination numbers of $$C_m$$ bundles over a cycle $$C_n$$Cn

Let \(G=(V,E)\) be a simple graph without isolated vertices. 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 \(S\). A paired dominating set of \(G\) is a dominating set whose induced subgraph has a perfect matching. The minimum cardinality of a total dominating set (respectively, a paired dominating set) is the total domination number (respectively, the paired domination number). Hu and Xu (J Combin Optim 27(2):369–378, 2014) computed the exact values of total and paired domination numbers of Cartesian product \(C_n\square C_m\) for \(m=3,4\). Graph bundles generalize the notions of covering graphs and Cartesian products. In this paper, we generalize these results given in Hu and Xu (J Combin Optim 27(2):369–378, 2014) to graph bundle and compute the total domination number and the paired domination number of \(C_m\) bundles over a cycle \(C_n\) for \(m=3,4\). Moreover, we give the exact value for the total domination number of Cartesian product \(C_n\square C_5\) and some upper bounds of \(C_m\) bundles over a cycle \(C_n\) where \(m\ge 5\).     

F_{3}-domination problem of graphs

Given a graph \(G\) and a set \(S\subseteq V(G),\) a vertex \(v\) is said to be \(F_{3}\) -dominated by a vertex \(w\) in \(S\) if either \(v=w,\) or \(v\notin S\) and there exists a vertex \(u\) in \(V(G)-S\) such that \(P:wuv\) is a path in \(G\) . A set \(S\subseteq V(G)\) is an \(F_{3}\) -dominating set of \(G\) if every vertex \(v\) is \(F_{3}\) -dominated by a vertex \(w\) in \(S.\) The \(F_{3}\) -domination number of \(G\) , denoted by \(\gamma _{F_{3}}(G)\) , is the minimum cardinality of an \(F_{3}\) -dominating set of \(G\) . In this paper, we study the \(F_{3}\) -domination of Cartesian product of graphs, and give formulas to compute the \(F_{3}\) -domination number of \(P_{m}\times P_{n}\) and \(P_{m}\times C_{n}\) for special \(m,n.\)      

On Split-Coloring Problems

We study a new coloring concept which generalizes the classical vertex coloring problem in a graph by extending the notion of stable sets to split graphs. First of all, we propose the packing problem of finding the split graph of maximum size where a split graph is a graph G = (V,E) in which the vertex set V can be partitioned into a clique K and a stable set S. No condition is imposed on the edges linking vertices in S to the vertices in K. This maximum split graph problem gives rise to an associated partitioning problem that we call the split-coloring problem. Given a graph, the objective is to cover all his vertices by a least number of split graphs. Definitions related to this new problem are introduced. We mention some polynomially solvable cases and describe open questions on this area. An erratum to this article is available at .