Path connectivity of line graphs

Dirac showed that in a $(k-1)$-connected graph there is a path through each $k$ vertices. The path $k$-connectivity $\pi_k(G)$ of a graph $G$, which is a generalization of Dirac's notion, was introduced by Hager in 1986. In this paper, we study path connectivity of line graphs.Comment: 15 pages. arXiv admin note: substantial text overlap with arXiv:1508.07202, arXiv:1207.1838; text overlap with arXiv:1103.6095 by other author

Pendant-tree connectivity of line graphs

The concept of pendant-tree connectivity, introduced by Hager in 1985, is a generalization of classical vertex-connectivity. In this paper, we study pendant-tree connectivity of line graphs.Comment: 19 pagers, 2 figures. arXiv admin note: substantial text overlap with arXiv:1603.03995, arXiv:1508.07202, arXiv:1508.07149. text overlap with arXiv:1103.6095 by other author

A survey on the generalized connectivity of graphs

The generalized $k$-connectivity $\kappa_k(G)$ of a graph $G$ was introduced by Hager before 1985. As its a natural counterpart, we introduced the concept of generalized edge-connectivity $\lambda_k(G)$, recently. In this paper we summarize the known results on the generalized connectivity and generalized edge-connectivity. After an introductory section, the paper is then divided into nine sections: the generalized (edge-)connectivity of some graph classes, algorithms and computational complexity, sharp bounds of $\kappa_k(G)$ and $\lambda_k(G)$, graphs with large generalized (edge-)connectivity, Nordhaus-Gaddum-type results, graph operations, extremal problems, and some results for random graphs and multigraphs. It also contains some conjectures and open problems for further studies.Comment: 51 pages. arXiv admin note: text overlap with arXiv:1303.3881 by other author

Online Degree-Bounded Steiner Network Design

We initiate the study of degree-bounded network design problems in the online setting. The degree-bounded Steiner tree problem { which asks for a subgraph with minimum degree that connects a given set of vertices { is perhaps one of the most representative problems in this class. This paper deals with its well-studied generalization called the degree-bounded Steiner forest problem where the connectivity demands are represented by vertex pairs that need to be individually connected. In the classical online model, the input graph is given online but the demand pairs arrive sequentially in online steps. The selected subgraph starts off as the empty subgraph, but has to be augmented to satisfy the new connectivity constraint in each online step. The goal is to be competitive against an adversary that knows the input in advance. We design a simple greedy-like algorithm that achieves a competitive ratio of O(log n) where n is the number of vertices. We show that no (randomized) algorithm can achieve a (multiplicative) competitive ratio o(log n); thus our result is asymptotically tight. We further show strong hardness results for the group Steiner tree and the edge-weighted variants of degree-bounded connectivity problems. Fourer and Raghavachari resolved the online variant of degree-bounded Steiner forest in their paper in SODA'92. Since then, the natural family of degree-bounded network design problems has been extensively studied in the literature resulting in the development of many interesting tools and numerous papers on the topic. We hope that our approach in this paper, paves the way for solving the online variants of the classical problems in this family of network design problems

Solving a problem of angiogenesis of degree three

An absorbing weighted Fermat-Torricelli tree of degree four is a weighted Fermat-Torricelli tree of degree four which is derived as a limiting tree structure from a generalized Gauss tree of degree three (weighted full Steiner tree) of the same boundary convex quadrilateral in R^2: By letting the four variable positive weights which correspond to the fixed vertices of the quadrilateral and satisfy the dynamic plasticity equations of the weighted quadrilateral, we obtain a family of limiting tree structures of generalized Gauss trees which concentrate to the same weighted Fermat-Torricelli tree of degree four (universal absorbing Fermat-Torricelli tree). The values of the residual absorbing rates for each derived weighted Fermat-Torricelli tree of degree four of the universal Fermat-Torricelli tree form a universal absorbing set. The minimum of the universal absorbing Fermat-Torricelli set is responsible for the creation of a generalized Gauss tree of degree three for a boundary convex quadrilateral derived by a weighted Fermat-Torricelli tree of a boundary triangle (Angiogenesis of degree three). Each value from the universal absorbing set contains an evolutionary process of a generalized Gauss tree of degree three.Comment: 21 pages. 8 figures, Submitted to a Journa

The plasticity of non-overlapping convex sets in R^{2}

We study a generalization of the weighted Fermat-Torricelli problem in the plane, which is derived by replacing vertices of a convex polygon by 'small' closed convex curves with weights being positive real numbers on the curves, we also study its generalized inverse problem. Our solution of the problems is based on the first variation formula of the length of line segments that connect the weighted Fermat-Torricelli point with its projections onto given closed convex curves. We find the 'plasticity' solutions for non-overlapping circles with variable radius.Comment: 11 pages, 1 figure, submitted to a journa

Online Weighted Degree-Bounded Steiner Networks via Novel Online Mixed Packing/Covering

We design the first online algorithm with poly-logarithmic competitive ratio for the edge-weighted degree-bounded Steiner forest(EW-DB-SF) problem and its generalized variant. We obtain our result by demonstrating a new generic approach for solving mixed packing/covering integer programs in the online paradigm. In EW-DB-SF we are given an edge-weighted graph with a degree bound for every vertex. Given a root vertex in advance we receive a sequence of terminal vertices in an online manner. Upon the arrival of a terminal we need to augment our solution subgraph to connect the new terminal to the root. The goal is to minimize the total weight of the solution while respecting the degree bounds on the vertices. In the offline setting edge-weighted degree-bounded Steiner tree (EW-DB-ST) and its many variations have been extensively studied since early eighties. Unfortunately the recent advancements in the online network design problems are inherently difficult to adapt for degree-bounded problems. In contrast in this paper we obtain our result by using structural properties of the optimal solution, and reducing the EW-DB-SF problem to an exponential-size mixed packing/covering integer program in which every variable appears only once in covering constraints. We then design a generic integral algorithm for solving this restricted family of IPs. We demonstrate a new technique for solving mixed packing/covering integer programs. Define the covering frequency k of a program as the maximum number of covering constraints in which a variable can participate. Let m denote the number of packing constraints. We design an online deterministic integral algorithm with competitive ratio of O(k log m) for the mixed packing/covering integer programs. We believe this technique can be used as an interesting alternative for the standard primal-dual techniques in solving online problems

Optimal Networks

This mini-course was given in the First Yaroslavl Summer School on Discrete and Computational Geometry in August 2012, organized by International Delaunay Laboratory "Discrete and Computational Geometry" of Demidov Yaroslavl State University. The aim of this mini-course is to give an introduction in Optimal Networks theory. Optimal networks appear as solutions of the following natural problem: How to connect a finite set of points in a metric space in an optimal way? We cover three most natural types of optimal connection: spanning trees connection without additional road forks, shortest trees and locally shortest trees, and minimal fillings

Analytic Deformations of Minimal Networks

A behavior of extreme networks under deformations of their boundary sets is investigated. It is shown that analyticity of a deformation of boundary set guarantees preservation of the networks types for minimal spanning trees, minimal fillings and so-called stable shortest trees in the Euclidean space.Comment: 20 pages, 2 figure

Applications of variational analysis to a generalized Fermat-Torricelli problem

In this paper we develop new applications of variational analysis and generalized differentiation to the following optimization problem and its specifications: given n closed subsets of a Banach space, find such a point for which the sum of its distances to these sets is minimal. This problem can be viewed as an extension of the celebrated Fermat-Torricelli problem: given three points on the plane, find another point such that the sum of its distances to the designated points is minimal. The generalized Fermat-Torricelli problem formulated and studied in this paper is of undoubted mathematical interest and is promising for various applications including those frequently arising in location science, optimal networks, etc. Based on advanced tools and recent results of variational analysis and generalized differentiation, we derive necessary as well as necessary and sufficient optimality conditions for the extended version of the Fermat-Torricelli problem under consideration, which allow us to completely solve it in some important settings. Furthermore, we develop and justify a numerical algorithm of the subgradient type to find optimal solutions in convex settings and provide its numerical implementations