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.     

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.     

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...     

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.     

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.     

Co-2-plex polynomials

This paper offers a generalization of the independence polynomial, the co-k-plex polynomial. The resulting family of polynomials carries combinatorial information on a class of independence systems defined over the vertex set of a finite graph. Specifically, we offer a recursion formula and examples of the co-2-plex polynomial on certain graphs. In addition, we characterize the class of graphs whose co-2-plex polynomial will have all real roots.     

Vertex and Tree Arboricities of Graphs

This paper studies the following variations of arboricity of graphs. The vertex (respectively, tree) arboricity of a graph G is the minimum number va(G) (respectively, ta(G)) of subsets into which the vertices of G can be partitioned so that each subset induces a forest (respectively, tree). This paper studies the vertex and the tree arboricities on various classes of graphs for exact values, algorithms, bounds, hamiltonicity and NP-completeness. The graphs investigated in this paper include block-cactus graphs, series-parallel graphs, cographs and planar graphs.     

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 .     

Algebraic connectivity of an even uniform hypergraph

We generalize Laplacian matrices for graphs to Laplacian tensors for even uniform hypergraphs and set some foundations for the spectral hypergraph theory based upon Laplacian tensors. Especially, algebraic connectivity of an even uniform hypergraph based on Z-eigenvalues of the corresponding Laplacian tensor is introduced and its connections with edge connectivity and vertex connectivity are discussed.     

Modified linear programming and class 0 bounds for graph pebbling

Given a configuration of pebbles on the vertices of a connected graph G, a pebbling move removes two pebbles from some vertex and places one pebble on an adjacent vertex. The pebbling number of a graph G is the smallest integer k such that for each vertex v and each configuration of k pebbles on G there is a sequence of pebbling moves that places at least one pebble on v. First, we improve on results of Hurlbert, who introduced a linear optimization technique for graph pebbling. In particular, we use a different set of weight functions, based on graphs more general than trees. We apply this new idea to some graphs from Hurlbert’s paper to give improved bounds on their pebbling numbers. Second, we investigate the structure of Class 0 graphs with few edges. We show that every n-vertex Class 0 graph has at least \(\frac{5}{3}n - \frac{11}{3}\) edges. This disproves a conjecture of Blasiak et al. For diameter 2 graphs, we strengthen this lower bound to \(2n - 5\), which is best possible. Further, we characterize the graphs where the bound holds with equality and extend the argument to obtain an identical bound for diameter 2 graphs with no cut-vertex.     

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.     

On computing a minimum secure dominating set in block graphs

In a graph \(G=(V,E)\), a set \(D \subseteq V\) is said to be a dominating set of G if for every vertex \(u\in V{\setminus }D\), there exists a vertex \(v\in D\) such that \(uv\in E\). A secure dominating set of the graph G is a dominating set D of G such that for every \(u\in V{\setminus }D\), there exists a vertex \(v\in D\) such that \(uv\in E\) and \((D{\setminus }\{v\})\cup \{u\}\) is a dominating set of G. Given a graph G and a positive integer k, the secure domination problem is to decide whether G has a secure dominating set of cardinality at most k. The secure domination problem has been shown to be NP-complete for chordal graphs via split graphs and for bipartite graphs. In Liu et al. (in: Proceedings of 27th workshop on combinatorial mathematics and computation theory, 2010), it is asked to find a polynomial time algorithm for computing a minimum secure dominating set in a block graph. In this paper, we answer this by presenting a linear time algorithm to compute a minimum secure dominating set in block graphs. We then strengthen the known NP-completeness of the secure domination problem by showing that the secure domination problem is NP-complete for undirected path graphs and chordal bipartite graphs.     

Penta-Extensions of Hereditary Classes of Graphs

Penta is the configuration shown in figure 1(a), where continuous lines represent edges and dotted lines represent non-edges. The vertex u in figure 1(a) is called the center of Penta. A graph G is called a pentagraph if every induced subgraph H of G has a vertex v which is not a center of induced Penta in H. The class of pentagraphs is a common generalization of chordal [triangulated] graphs and Mahadev graphs. We construct a polynomial-time algorithm that either find a maximum stable set of G or concludes that G is not a pentagraph. We propose a method for extending α-polynomial hereditary classes based on induced Pentas.     

Acyclically 3-colorable planar graphs

In this paper we study the acyclic 3-colorability of some subclasses of planar graphs. First, we show that there exist infinite classes of cubic planar graphs that are not acyclically 3-colorable. Then, we show that every planar graph has a subdivision with one vertex per edge that is acyclically 3-colorable and provide a linear-time coloring algorithm. Finally, we characterize the series-parallel graphs for which every 3-coloring is acyclic and provide a linear-time recognition algorithm for such graphs.     

A PTAS for the minimum weighted dominating set problem with smooth weights on unit disk graphs

In the minimum weighted dominating set problem (MWDS), we are given a unit disk graph with non-negative weight on each vertex. The MWDS seeks a subset of the vertices of the graph with minimum total weight such that each vertex of the graph is either in the subset or adjacent to some nodes in the subset. A?weight function is called smooth, if the ratio of the weights of any two adjacent nodes is upper bounded by a constant. MWDS is known to be NP-hard. In this paper, we give the first polynomial time approximation scheme (PTAS) for MWDS with smooth weights on unit disk graphs, which achieves a (1+ε)-approximation for MWDS, for any ε>0.     

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.     

Broadcasting multiple messages in the 1-in port model in optimal time

In the 1-in port model, every vertex of a synchronous network can receive at most one message in each time unit. We consider simultaneous broadcasting of multiple messages from the same source or from distinct sources in such networks with an additional restriction that every received message can be sent out to neighbors only in the next time unit and never to already informed vertex. We use a general concept of level-disjoint partitions developed for this scenario. Here we introduce a subgraph extension technique for efficient spreading information within this concept. Surprisingly, this approach with so called biwheels leads to simultaneous broadcasting of optimal number of messages on a wide class of graphs in optimal time. In particular, we provide tight results for bipartite tori, meshes, hypercubes, Knödel graphs, circulant graphs. We also propose several open problems and conjectures.     

Minimum statuses of connected graphs with fixed maximum degree and order

The status of a vertex in a connected graph is the sum of distances between the vertex and all vertices. The minimum status of a connected graph is the minimum of statuses of all vertices of this graph. In this paper we obtain the sharp lower bound and the sharp upper bound on the minimum status of a connected graph with maximum degree k and order n. All the graphs attaining the lower bound are obtained, and a necessary condition is given for those graphs attaining the upper bound.     

Algorithm complexity of neighborhood total domination and $$(\rho ,\gamma _{nt})$$-graphs

A neighborhood total dominating set, abbreviated for NTD-set D, is a vertex set of G such that D is a dominating set with an extra property: the subgraph induced by the open neighborhood of D has no isolated vertex. The neighborhood total domination number, denoted by \(\gamma _{nt}(G)\), is the minimum cardinality of a NTD-set in G. In this paper, we prove that NTD problem is NP-complete for bipartite graphs and split graphs. Then we give a linear-time algorithm to determine \(\gamma _{nt}(T)\) for a given tree T. Finally, we characterize a constructive property of \((\gamma _{nt},2\gamma )\)-trees and provide a constructive characterization for \((\rho ,\gamma _{nt})\)-graphs, where \(\gamma \) and \(\rho \) are domination number and packing number for the given graph, respectively.     

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.