Let G = (V, E) be a graph and w: E -» R+ be a length function. Given S C V, a Steiner tour is a cycle passing at least once through each vertex of S . In this paper we investigate naturally submodular graphs: graphs for which the length function of the Steiner tours is submodular. We provide two characterizations of naturally submodular graphs, an 0(n) time algorithm for identifying such graphs...

Given a directed graph G and a list (s1, t1), . . . , (sk, tk) of terminal pairs, the Directed Steiner Network problem asks for a minimum-cost subgraph of G that contains a directed si → ti path for every 1 ≤ i ≤ k. The special case Directed Steiner Tree (when we ask for paths from a root r to terminals t1, . . . , tk) is known to be fixed-parameter tractable parameterized by the number of term...

the wiener index w(g) of a connected graph g is defined as the sum of the distances betweenall unordered pairs of vertices of g. the eccentricity of a vertex v in g is the distance to avertex farthest from v. in this paper we obtain the wiener index of a graph in terms ofeccentricities. further we extend these results to the self-centered graphs.

Given an undirected hypergraph and a subset of vertices S ⊆ V with a specified root vertex r ∈ S, the Steiner Rooted-Orientation problem is to find an orientation of all the hyperedges so that in the resulting directed hypergraph the “connectivity” from the root r to the vertices in S is maximized. This is motivated by a multicasting problem in undirected networks as well as a generalization of...

A cubic graph G is S-edge-colorable for a Steiner triple system S if its edges can be colored with points of S in such a way that the points assigned to three edges Electronic Notes in Discrete Mathematics 29 (2007) 23–27 1571-0653/$ – see front matter © 2007 Elsevier B.V. All rights reserved. www.elsevier.com/locate/endm doi:10.1016/j.endm.2007.07.005 sharing a vertex form a triple in S. We sh...

The block intersection graph of a combinatorial design with block set B is the graph with B as its vertex set such that two vertices are adjacent if and only if their associated blocks are not disjoint. The chromatic index of a graph G is the least number of colours that enable each edge of G to be assigned a single colour such that adjacent edges never have the same colour. A graph G for which...

We show that for every admissible order v ≡ 0 or 2 (mod 6) there exists a near-Steiner triple system of order v that can be halved. As a corollary we obtain that a Steiner almost self-complementary graph with n vertices exists if and only if n ≡ 0 or 2 (mod 6). © 2008 Elsevier B.V. All rights reserved.

Steiner tree problem on the graph is an NPComplete problem and has no exact solution in polynomial time. Since this problem is practically useful, there are more attentions to heuristic and approximation approaches rather than exact ones. By using heuristic algorithms, the near optimum answers are obtained in polynomial time that this is faster than exact approaches. The goal of Steiner Tree pr...

In this paper we provide an approximation à la Ambrosio-Tortorelli of some classical minimization problems involving the length of an unknown one-dimensional set, with an additional connectedness constraint, in dimension two. We introduce a term of new type relying on a weighted geodesic distance that forces the minimizers to be connected at the limit. We apply this approach to approximate the ...