Bounding the total domination subdivision number of a graph in terms of its order

The total domination subdivision number \(\mathrm{sd}_{\gamma _{t}}(G)\) of a graph 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 \(\mathrm{sd}_{\gamma_{t}}(G)\leq \lfloor\frac{2n}{3}\rfloor\) for any simple connected graph G of order n≥3 other than K ₄. We also determine all simple connected graphs G with \(\mathrm{sd}_{\gamma_{t}}(G)=\lfloor\frac{2n}{3}\rfloor\).     

A short proof of a result on a Vizing-like problem for integer total domination

In this note, we provide a short proof of a recent result on a Vizing-like problem for integer total domination in graphs given by Li and Hou (J. Comb. Optim., 2008).     

Fibonacci index and stability number of graphs: a polyhedral study

The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory. In this paper, we establish tight upper bounds for the Fibonacci index in terms of the stability number and the order of general graphs and connected graphs. Turán graphs frequently appear in extremal graph theory. We show that Turán graphs and a connected variant of them are also extremal for these particular problems. We also make a polyhedral study by establishing all the optimal linear inequalities for the stability number and the Fibonacci index, inside the classes of general and connected graphs of order n.     

An effective algorithm for obtaining the whole set of minimal cost pairs of disjoint paths with dual arc costs

In telecommunication networks design the problem of obtaining optimal (arc or node) disjoint paths, for increasing network reliability, is extremely important. The problem of calculating k _c disjoint paths from s to t (two distinct nodes), in a network with k _c different (arbitrary) costs on every arc such that the total cost of the paths is minimised, is NP-complete even for k _c =2. When k _c =2 these networks are usually designated as dual arc cost networks.     

On sorting unsigned permutations by double-cut-and-joins

The problem of sorting unsigned permutations by double-cut-and-joins (SBD) arises when we perform the double-cut-and-join (DCJ) operations on pairs of unichromosomal genomes without the gene strandedness information. In this paper we show it is a NP-hard problem by reduction to an equivalent previously-known problem, called breakpoint graph decomposition (BGD), which calls for a largest collection of edge-disjoint alternating cycles in a breakpoint graph. To obtain a better approximation algorithm for the SBD problem, we made a suitable modification to Lin and Jiang’s algorithm which was initially proposed to approximate the BGD problem, and then carried out a rigorous performance analysis via fractional linear programming. The approximation ratio thus achieved for the SBD problem is $\frac{17}{12}+\epsilon \approx 1.4167 +\epsilon$ , for any positive ε.     

On dual power assignment optimization for biconnectivity

Topology control is an important technology of wireless ad hoc networks to achieve energy efficiency and fault tolerance. In this paper, we study the dual power assignment problem for 2-edge connectivity and 2-vertex connectivity in the symmetric graphical model which is a combinatorial optimization problem from topology control technology.The problem is arisen from the following origin. In a wireless ad hoc network where each node can switch its transmission power between high-level and low-level, how can we establish a fault-tolerantly connected network topology in the most energy-efficient way? Specifically, the objective is to minimize the number of nodes assigned with high power and yet achieve 2-edge connectivity or 2-vertex connectivity.We addressed these optimization problems (2-edge connectivity and 2-vertex connectivity version) under the general graph model in (Wang et al. in Theor. Comput. Sci., 2008). In this paper, we propose a novel approximation algorithm, called Candidate Set Filtering algorithm, to compute nearly-optimal solutions. Specifically, our algorithm can achieve 3.67-approximation ratio for both 2-edge connectivity and 2-vertex connectivity, which improves the existing 4-approximation algorithms for these two cases.     

Half integer extreme points in the linear relaxation of the 2-edge-connected subgraph polyhedron

This paper studies the graphs for which the linear relaxation of the 2-connected spanning subgraph polyhedron has integer or half-integer extreme points. These graphs are called quasi-integer. For these graphs, the linear relaxation of the k-edge connected spanning subgraph polyhedron is integer for all k=4r, r≥1. The class of quasi-integer graphs is closed under minors and contains for instance the class of series-parallel graphs. We discuss some structural properties of graphs which are minimally non quasi-integer graphs, then we examine some basic operations which preserve the quasi-integer property. Using this, we show that the subdivisions of wheels are quasi-integer.     

Integration capabilities as mediator of product development practices–performance

The use of integration practices, both internal (where various functions work together) and external (links with customers and suppliers during development), are espoused in the new product development (NPD) literature. However, empirical findings in the literature suggest adoption of integration practices does not necessarily lead to positive performance. We introduce the concept of integration capabilities to explain the relationship between use of integration practices and NPD performance. We tested a mediation model using data from 141 Japanese and American firms and found that effects of both types of integration on time and product performance were mediated by the integration capabilities developed. We also found differential effects of the type of integration. The findings demonstrate that developing superior integration capabilities are needed for companies to meet and exceed product development expectations in terms of both product and time performance. Simply, a company may utilize integration practices but if it does not utilize them in such a way as to generate real capabilities, the use of integration practices may not lead to positive performance effects.     

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.     

Long cycles in hypercubes with optimal number of?faulty vertices

Let f(n) be the maximum integer such that for every set F of at most f(n) vertices of the hypercube Q _n , there exists a cycle of length at least 2 ⁿ ?2|F| in Q _n ?F. Casta?eda and Gotchev conjectured that $f(n)=\binom{n}{2}-2$ . We prove this conjecture. We also prove that for every set F of at most (n ²+n?4)/4 vertices of Q _n , there exists a path of length at least 2 ⁿ ?2|F|?2 in Q _n ?F between any two vertices such that each of them has at most 3 neighbors in F. We introduce a new technique of potentials which could be of independent interest.     

Linear time construction of 5-phylogenetic roots for tree chordal graphs

Inspired by phylogenetic tree construction in computational biology, Lin et al. (The 11th Annual International Symposium on Algorithms and Computation (ISAAC 2000), pp. 539–551, 2000) introduced the notion of a k -phylogenetic root. A k-phylogenetic root of a graph G is a tree T such that the leaves of T are the vertices of G, two vertices are adjacent in G precisely if they are within distance k in T, and all non-leaf vertices of T have degree at least three. The k-phylogenetic root problem is to decide whether such a tree T exists for a given graph G. In addition to introducing this problem, Lin et al. designed linear time constructive algorithms for k≤4, while left the problem open for k≥5. In this paper, we partially fill this hole by giving a linear time constructive algorithm to decide whether a given tree chordal graph has a 5-phylogenetic root; this is the largest class of graphs known to have such a construction.     

Error-correcting pooling designs associated with the dual space of unitary space and ratio efficiency comparison

In this paper, we construct a d ^z -disjunct matrix with subspaces in a dual space of Unitary space , then give its several properties. As the smaller the ratio efficiency is, the better the pooling design is. We compare the ratio efficiency of this construction with others, such as the ratio efficiency of the construction of set, the general space and the dual space of symplectic space. In addition, we find it smaller under some conditions. Supported by NSF of the Education Department of Hebei Province (2007127) and NSF of Hebei Normal University (L2004B04).     

Against the Antagonist View of Professional – Manager Relationships: The Case of the Culture Industry

In the emerging knowledge society, a major challenge for human resource management theory and practice is how to lead professionals and experts in day-to-day work. The literature on professionals suggests that the relationship between professionals and managers is complicated, as professional ideologies, practices and interests tend to clash with organizational and managerial objectives. This paper questions such an antagonist view and suggests that the relationship between the two groups is a rather intricate one and that there is no such clear disjunction between the two groups. In addition, there is a strong reliance on cases from the health care sector, especially in an Anglo-American context. The paper reports a study of the relationship between professionals, officers and politicians in the Swedish culture sector in the largest Swedish region. The findings suggest that professionals and managers are capable of taking the role of the other and see a broader perspective than merely individual interests.     

Nordhaus–Gaddum type result for the matching number of a graph

For a graph G, \(\alpha '(G)\) is the matching number of G. Let \(k\ge 2\) be an integer, \(K_{n}\) be the complete graph of order n. Assume that \(G_{1}, G_{2}, \ldots , G_{k}\) is a k-decomposition of \(K_{n}\). In this paper, we show that (1)

$$\begin{aligned} \left\lfloor \frac{n}{2}\right\rfloor \le \sum _{i=1}^{k} \alpha '(G_{i})\le k\left\lfloor \frac{n}{2}\right\rfloor . \end{aligned}$$

(2) If each \(G_{i}\) is non-empty for \(i = 1, \ldots , k\), then for \(n\ge 6k\),

$$\begin{aligned} \sum _{i=1}^{k} \alpha '(G_{i})\ge \left\lfloor \frac{n+k-1}{2}\right\rfloor . \end{aligned}$$

(3) If \(G_{i}\) has no isolated vertices for \(i = 1, \ldots , k\), then for \(n\ge 8k\),

$$\begin{aligned} \sum _{i=1}^{k} \alpha '(G_{i})\ge \left\lfloor \frac{n}{2}\right\rfloor +k. \end{aligned}$$

The bounds in (1), (2) and (3) are sharp. (4) When \(k= 2\), we characterize all the extremal graphs which attain the lower bounds in (1), (2) and (3), respectively.

     

Improved floor-planning of graphs via adjacency-preserving transformations

Let G=(V,E) and G′=(V′,E′) be two graphs, an adjacency-preserving transformation from G to G′ is a one-to-many and onto mapping from V to V′ satisfying the following: (1) Each vertex v∈V in G is mapped to a non-empty subset \(\mathcal{A}(v)\subset V'\) in G′. The subgraph induced by \(\mathcal{A}(v)\) is a connected subgraph of G′; (2) if u≠v∈V, then \(\mathcal{A}(u)\cap\mathcal{A}(v)=\emptyset\); and (3) two vertices u and v are adjacent to each other in G if and only if subgraphs induced by \(\mathcal{A}(u)\) and \(\mathcal{A}(v)\) are connected in G′.In this paper, we study adjacency-preserving transformations from plane triangulations to irreducible triangulations (which are internally triangulated, with four exterior vertices and no separating triangles). As one shall see, our transformations not only preserve adjacency well, but also preserve the endowed realizers of plane triangulations well in the endowed transversal structures of the image irreducible triangulations, which may be desirable in some applications.We then present such an application in floor-planning of plane graphs. The expected grid size of the floor-plan of our linear time algorithm is improved to \((\frac{5n}{8}+O(1))\times (\frac{23n}{24}+O(1))\), though the worst case grid size bound of the algorithm remains \(\lfloor\frac{2n+1}{3}\rfloor\times(n-1)\), which is the same as the algorithm presented in Liao et al. (J. Algorithms 48:441–451, 2003).     

The competition number of a graph with?exactly?two?holes

Given an acyclic digraph D, the competition graph C(D) of D is the graph with the same vertex set as D and two distinct vertices x and y are adjacent in C(D) if and only if there is a vertex v in D such that (x,v) and (y,v) are arcs of D. The competition number κ(G) of a graph G is the least number of isolated vertices that must be added to G to form a competition graph. The purpose of this paper is to prove that the competition number of a graph with exactly two holes is at most three.     

Fixed-parameter tractability of anonymizing data by suppressing entries

A popular model for protecting privacy when person-specific data is released is k -anonymity. A dataset is k-anonymous if each record is identical to at least (k−1) other records in the dataset. The basic k-anonymization problem, which minimizes the number of dataset entries that must be suppressed to achieve k-anonymity, is NP-hard and hence not solvable both quickly and optimally in general. We apply parameterized complexity analysis to explore algorithmic options for restricted versions of this problem that occur in practice. We present the first fixed-parameter algorithms for this problem and identify key techniques that can be applied to this and other k-anonymization problems.     

How do actors shape social networks during the process of new product development?

Recent studies on new-idea generation and development have highlighted the role played by network structure in the genesis of new combinations or the process of selecting ideas. However, less attention has been paid to the factors that entice actors to shape social networks during the process of the development of new ideas.This research was conducted in an R&D facility of a semi-conductor company. We analysed the generation of five creative projects and their development over a four-year period. We used a longitudinal approach and collected data through interviews and observations to identify the creative contributions and the actors who were involved at different time periods for each project. We mapped the relationships between actors who contributed to the development of each idea through creative thinking and/or helped it to become accepted both internally and externally over three-year windows. This method generated data on network evolution.We also carried out a qualitative analysis and identified four main factors explaining why actors turn to others during the idea-development process: (1) to gain access to information; (2) to enhance credibility; (3) to exercise one’s influence; and (4) to gain access to knowledge through people or objects. We demonstrate that different types of ties or network structures are relied upon to reap different kinds of benefits. This may partially explain network evolution as an idea progresses through different development stages.     

Algebraic and combinatorial properties of ideals and algebras of uniform clutters of TDI systems

Let \(\mathcal{C}\) be a uniform clutter and let A be the incidence matrix of \(\mathcal{C}\). We denote the column vectors of A by v ₁,…,v _q . Under certain conditions we prove that \(\mathcal{C}\) is vertex critical. If \(\mathcal{C}\) satisfies the max-flow min-cut property, we prove that A diagonalizes over ? to an identity matrix and that v ₁,…,v _q form a Hilbert basis. We also prove that if \(\mathcal{C}\) has a perfect matching such that \(\mathcal{C}\) has the packing property and its vertex covering number is equal to 2, then A diagonalizes over ? to an identity matrix. If A is a balanced matrix we prove that any regular triangulation of the cone generated by v ₁,…,v _q is unimodular. Some examples are presented to show that our results only hold for uniform clutters. These results are closely related to certain algebraic properties, such as the normality or torsion-freeness, of blowup algebras of edge ideals and to finitely generated abelian groups. They are also related to the theory of Gröbner bases of toric ideals and to Ehrhart rings.