The domination number of graph is the smallest cardinality set G. A subset a vertex S G called if every element dominates G, meaning that not an connected and one distance from S. has become interesting research studies on several graphs k -connected such as circulant graphs, grids, wheels. This study aims to determine other k-connected Harary graph. method used pattern detection axiomatic dedu...

For a real number p ≥ 2, an integer k > 0 and a set of terminals X in the plane, the Euclidean power-p Steiner tree problem asks for a tree interconnecting X and at most k Steiner points such that the sum of the p-th powers of the edge lengths is minimised. We show that this problem is in the complexity subclass exp-APX (but not poly-APX) of NPO. We then demonstrate that the approximation algor...

Given a complete graph G = (V,E), where each vertex is labeled either terminal or Steiner, a distance function d : E → R, and a positive integer k, we study the problem of finding a Steiner tree T spanning all terminals and at most k Steiner vertices, such that the length of the longest edge is minimized. We first show that this problem is NP-hard and cannot be approximated within a factor 2− ε...

We extend the primal-dual approximation technique of Goemans and Williamson to the Steiner connectivity problem, a kind of Steiner tree problem in hypergraphs. This yields a (k+1)-approximation algorithm for the case that k is the minimum of the maximal number of nodes in a hyperedge minus 1 and the maximal number of terminal nodes in a hyperedge. These results require the proof of a degree pro...

The geodesic and geodesic interval, namely the set of all vertices lying on geodesics between a pair of vertices in a connected graph, is a part of folklore in metric graph theory. It is also known that Steiner tree of a (multi) set with k (k > 2) vertices, generalizes geodesics. In [1] the authors studied the k-Steiner intervals S(u1, u2, . . . , uk) on connected graphs (k ≥ 3) as the k-ary ge...

25 The Traveling Salesman Problem (TSP) is still one of the most researched topics in computational mathematics, and we introduce a variant of it, namely the study of the closed k-walks in graphs. We 2 G. Bullington, R. Gera, L. Eroh And S.J. Winters search for a shortest closed route visiting k cities in a non complete graph without weights. This motivates the following definition. Given a set...

Given a properly face two-coloured triangulation of the graph Kn in a surface, a Steiner triple system can be constructed from each of the colour classes. The two Steiner triple systems obtained in this manner are said to form a biembedding. If the systems are isomorphic to each other it is a self-embedding. In the following, for each k ≥ 2, we construct a self-embedding of the doubled affine S...

The problem of minimum cost in-network fusion of measurements, collected from distributed sensors via multihop routing is considered. A designated fusion center collects all the sensor measurements and performs a statistical-inference test on the measurements, which are spatially correlated according to a Markov random field. Conditioned on the delivery of a sufficient statistic to the fusion c...

We introduce the concept of a clique bitrade, which generalizes several known types of bitrades, including latin bitrades, Steiner T(k−1, k, v) bitrades, extended 1perfect bitrades. For a distance-regular graph, we show a one-to-one correspondence between the clique bitrades that meet the weight-distribution lower bound on the cardinality and the bipartite isometric subgraphs that are distance-...