Properties of ℛ-Sausages

This paper is concerned with the Steiner ratio. A number of properties about the structure of the flat sausage and -Sausage convex polytopes yielding the best Steiner ratio in two- and three-dimensional Euclidean space, and the topology of the Steiner Minimal Tree for the corresponding vertex sets, are presented.     

Computing the metric dimension of convex polytopes generated by wheel related graphs

An ordered set \({W = \{w_1, . . . ,w_k\} \subseteqq V (G)}\) of vertices of G is called a resolving set or locating set for G if every vertex is uniquely determined by its vector of distances to the vertices in W. A resolving set of minimum cardinality is called a metric basis for G and this cardinality is the metric dimension or location number of G, denoted by \({\beta(G)}\).     

The Local Metric Dimension of Strong Product Graphs

A vertex \(v\in V(G)\) is said to distinguish two vertices \(x,y\in V(G)\) of a nontrivial connected graph G if the distance from v to x is different from the distance from v to y. A set \(S\subset V(G)\) is a local metric generator for G if every two adjacent vertices of G are distinguished by some vertex of S. A local metric generator with the minimum cardinality is called a local metric basis for G and its cardinality, the local metric dimension of G. It is known that the problem of computing the local metric dimension of a graph is NP-Complete. In this paper we study the problem of finding exact values or bounds for the local metric dimension of strong product of graphs.     

A partition-based relaxation for Steiner trees

The Steiner tree problem is a classical NP-hard optimization problem with a wide range of practical applications. In an instance of this problem, we are given an undirected graph G = (V, E), a set of terminals \({R\subseteq V}\) , and non-negative costs c _e for all edges \({e \in E}\) . Any tree that contains all terminals is called a Steiner tree; the goal is to find a minimum-cost Steiner tree. The vertices \({V \backslash R}\) are called Steiner vertices. The best approximation algorithm known for the Steiner tree problem is a greedy algorithm due to Robins and Zelikovsky (SIAM J Discrete Math 19(1):122–134, 2005); it achieves a performance guarantee of \({1+\frac{\ln 3}{2}\approx 1.55}\) . The best known linear programming (LP)-based algorithm, on the other hand, is due to Goemans and Bertsimas (Math Program 60:145–166, 1993) and achieves an approximation ratio of 2?2/|R|. In this paper we establish a link between greedy and LP-based approaches by showing that Robins and Zelikovsky’s algorithm can be viewed as an iterated primal-dual algorithm with respect to a novel LP relaxation. The LP used in the first iteration is stronger than the well-known bidirected cut relaxation. An instance is b-quasi-bipartite if each connected component of \({G \backslash R}\) has at most b vertices. We show that Robins’ and Zelikovsky’s algorithm has an approximation ratio better than \({1+\frac{\ln 3}{2}}\) for such instances, and we prove that the integrality gap of our LP is between \({\frac{8}{7}}\) and \({\frac{2b+1}{b+1}}\) .     

Isometric embeddings of finite metric spaces

It is proved that there exists a metric on a Cantor set such that any finite metric space whose diameter does not exceed 1 and the number of points does not exceed n can be isometrically embedded into it. It is also proved that for any m, n ∈ N there exists a Cantor set in R^m that isometrically contains all finite metric spaces which can be embedded into R^m, contain at most n points, and have the diameter at most 1. The latter result is proved for a wide class of metrics on R^m and, in particular, for the Euclidean metric.     

Comparing and Simplifying Distinct-Cluster Phylogenetic Networks

Phylogenetic networks are rooted acyclic directed graphs in which the leaves are identified with members of a set X of species. The cluster of a vertex is the set of leaves that are descendants of the vertex. A network is “distinct-cluster” if distinct vertices have distinct clusters. This paper focuses on the set DC(X) of distinct-cluster networks whose leaves are identified with the members of X. For a fixed X, a metric on DC(X) is defined. There is a “cluster-preserving” simplification process by which vertices or certain arcs may be removed without changing the clusters of any remaining vertices. Many of the resulting networks may be uniquely determined without regard to the order of the simplifying operations.     

The Steiner Mapping for Three Points in Euclidean Plane

For the Euclidean plane ? the Steiner mapping associating any three points a, b, c with their median s, and the corresponding operator P _D of metric projection of the space l ₁ ³ (?) onto its diagonal subspace D = {(x,x,x): x ∈ ?}, P _D (a,b,c) = (s,s,s): s are considered. The exact value of the linearity coefficient of P _D is calculated.     

Inner and outer approximation of convex sets using alignment

We show that there exists, for each closed bounded convex set C in the Euclidean plane with nonempty interior, a quadrangle Q having the following two properties. Its sides support C at the vertices of a rectangle r and at least three of the vertices of Q lie on the boundary of a rectangle R that is a dilation of r with ratio 2. We will prove that this implies that quadrangle Q is contained in rectangle R and that, consequently, the inner approximation r of C has an area of at least half the area of the outer approximation Q of C. The proof makes use of alignment or Schüttelung, an operation on convex sets.     

Super-Simple Twofold Steiner Pentagon Systems

A twofold pentagon system of order v is a decomposition of the complete undirected 2-multigraph 2K _v into pentagons. A twofold Steiner pentagon system of order v [TSPS(v)] is a twofold pentagon system such that every pair of distinct vertices is joined by a path of length two in exactly two pentagons of the system. A TSPS(v) is said to be super-simple if its underlying (v, 5, 4)-BIBD is super-simple; that is, if any two blocks of the BIBD intersect in at most two points. In this paper, it is shown that the necessary conditions for the existence of a super-simple TSPS(v); namely, v ≥ 15 and v ≡ 0 or 1 (mod 5) are sufficient. For these specified orders, the main result of this paper also guarantees the existence of a very special and interesting class of twofold and fourfold Steiner pentagon systems of order v with the additional property that, for any two vertices, the two or four paths of length two joining them are distinct.     

On the Metric Dimension of Imprimitive Distance-Regular Graphs

A resolving set for a graph \({\Gamma}\) is a collection of vertices S, chosen so that for each vertex v, the list of distances from v to the members of S uniquely specifies v. The metric dimension of \({\Gamma}\) is the smallest size of a resolving set for \({\Gamma}\). Much attention has been paid to the metric dimension of distance-regular graphs. Work of Babai from the early 1980s yields general bounds on the metric dimension of primitive distance-regular graphs in terms of their parameters. We show how the metric dimension of an imprimitive distance-regular graph can be related to that of its halved and folded graphs. We also consider infinite families (including Taylor graphs and the incidence graphs of certain symmetric designs) where more precise results are possible.     

Computational Complexity of the Vertex Cover Problem in the Class of Planar Triangulations

We study the computational complexity of the vertex cover problem in the class of planar graphs (planar triangulations) admitting a plane representation whose faces are triangles. It is shown that the problem is strongly NP-hard in the class of 4-connected planar triangulations in which the degrees of vertices are of order O(log n), where n is the number of vertices, and in the class of plane 4-connected Delaunay triangulations based on the Minkowski triangular distance. A pair of vertices in such a triangulation is adjacent if and only if there is an equilateral triangle ?(p, λ) with p ∈ R² and λ > 0 whose interior does not contain triangulation vertices and whose boundary contains this pair of vertices and only it, where ?(p, λ) = p + λ? = {x ∈ R²: x = p + λa, a ∈ ?}; here ? is the equilateral triangle with unit sides such that its barycenter is the origin and one of the vertices belongs to the negative y-axis. Keywords: computational complexity, Delaunay triangulation, Delaunay TD-triangulation.     

Strictly Deza line graphs

For a given graph G, its line graph L(G) is defined as the graph with vertex set equal to the edge set of G in which two vertices are adjacent if and only if the corresponding edges of G have exactly one common vertex. A k-regular graph of diameter 2 on υ vertices is called a strictly Deza graph with parameters (υ, k, b, a) if it is not strongly regular and any two vertices have a or b common neighbors. We give a classification of strictly Deza line graphs.     

The mapping taking three points of a Banach space to their Steiner point

A mapping St taking any three points a, b, c of a Banach space X into a set St(a,b,c) of their Steiner points and the corresponding operator P_D of metric projection of a space X × X × X onto its diagonal subspace D = {(x,x,x): x ∈ X}, P_D(a,b,c) = {(s,s,s): s ∈ St(a,b,c)}, are considered. The linearity coefficient of an arbitrary selection from P_D is estimated depending on properties of the space X. Estimates for the Lipschitz constant of an arbitrary selection from the mapping St are obtained as a corollary.     

Classification of amply regular graphs with <Emphasis Type="Italic">b</Emphasis><Subscript>1</Subscript> = 6

An undirected graph with v vertices in which the degrees of all vertices are equal to k, each edge is contained in exactly λ triangles, and the intersection of the neighborhoods of any two vertices at distance 2 contains exactly µ vertices is called amply regular with parameters (v, k, λ, µ). We complete the classification of amply regular graphs with b ₁ = 6, where b ₁ = k ? λ ? 1.     

Domination Defect in Graphs: Guarding With Fewer Guards

In this paper, we introduce a new graph parameter called the domination defect of a graph. The domination number γ of a graph G is the minimum number of vertices required to dominate the vertices of G. Due to the minimality of γ, if a set of vertices of G has cardinality less than γ then there are vertices of G that are not dominated by that set. The k-domination defect of G is the minimum number of vertices which are left un-dominated by a subset of γ - k vertices of G. We study different bounds on the k-domination defect of a graph G with respect to the domination number, order, degree sequence, graph homomorphisms and the existence of efficient dominating sets. We also characterize the graphs whose domination defect is 1 and find exact values of the domination defect for some particular classes of graphs.     

Independence numbers of random subgraphs of a distance graph

We consider the so-called distance graph G(n, 3, 1), whose vertices can be identified with three-element subsets of the set {1, 2,..., n}, two vertices being joined by an edge if and only if the corresponding subsets have exactly one common element. We study some properties of random subgraphs of G(n, 3, 1) in the Erd?s–Rényi model, in which each edge is included in the subgraph with some given probability p independently of the other edges. We find the asymptotics of the independence number of a random subgraph of G(n, 3, 1).     

On one test for the switching separability of graphs modulo <Emphasis Type="Italic">q</Emphasis>

We consider graphs whose edges are marked by numbers (weights) from 1 to q - 1 (with zero corresponding to the absence of an edge). A graph is additive if its vertices can be marked so that, for every two nonadjacent vertices, the sum of the marks modulo q is zero, and for adjacent vertices, it equals the weight of the corresponding edge. A switching of a given graph is its sum modulo q with some additive graph on the same set of vertices. A graph on n vertices is switching separable if some of its switchings has no connected components of size greater than n - 2. We consider the following separability test: If removing any vertex from G leads to a switching separable graph then G is switching separable. We prove this test for q odd and characterize the set of exclusions for q even. Connection is established between the switching separability of a graph and the reducibility of the n-ary quasigroup constructed from the graph.     

An Analog of Titchmarsh’s Theorem for the Fourier–Walsh Transform

The space clos(X) of all nonempty closed subsets of an unbounded metric space X is considered. The space clos(X) is endowed with a metric in which a sequence of closed sets converges if and only if the distances from these sets to a fixed point θ are bounded and, for any r, the sequence of the unions of the given sets with the exterior balls of radius r centered at θ converges in the Hausdorff metric. The metric on clos(X) thus defined is not equivalent to the Hausdorff metric, whatever the initial metric space X. Conditions for a set to be closed, totally bounded, or compact in clos(X) are obtained; criteria for the bounded compactness and separability of clos(X) are given. The space of continuous maps from a compact space to clos(X) is considered; conditions for a set to be totally bounded in this space are found.     

A sharp eigenvalue theorem for fractional elliptic equations

The invisibility graph I(X) of a set X ? R ^d is a (possibly infinite) graph whose vertices are the points of X and two vertices are connected by an edge if and only if the straight-line segment connecting the two corresponding points is not fully contained in X. We consider the following three parameters of a set X: the clique number ω(I(X)), the chromatic number χ(I(X)) and the convexity number γ(X), which is the minimum number of convex subsets of X that cover X.We settle a conjecture of Matou?ek and Valtr claiming that for every planar set X, γ(X) can be bounded in terms of χ(I(X)). As a part of the proof we show that a disc with n one-point holes near its boundary has χ(I(X)) ≥ log log(n) but ω(I(X)) = 3.We also find sets X in R⁵ with χ(X) = 2, but γ(X) arbitrarily large.     

An Optimal Algorithm for the <Emphasis Type="Italic">k</Emphasis>-Fixed-Endpoint Path Cover on Proper Interval Graphs

In this paper we consider the k-fixed-endpoint path cover problem on proper interval graphs, which is a generalization of the path cover problem. Given a graph G and a set T of k vertices, a k-fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint simple paths that covers the vertices of G, such that the vertices of T are all endpoints of these paths. The goal is to compute a k-fixed-endpoint path cover of G with minimum cardinality. We propose an optimal algorithm for this problem with runtime O(n), where n is the number of intervals in G. This algorithm is based on the Stair Normal Interval Representation (SNIR) matrix that characterizes proper interval graphs. In this characterization, every maximal clique of the graph is represented by one matrix element; the proposed algorithm uses this structural property, in order to determine directly the paths in an optimal solution.