Let G = (V,E) be a simple graph and let T = (P,B) be a Steiner triple system. Let φ be a one-to-one function from V to P . Any edge e = {u, v} has its image {φ(u), φ(v)} in a unique block in B. We also denote this induced function from edges to blocks by φ. We say that T represents G if there exists a one-to-one function φ : V → P such that the induced function φ : E → B is also one-to-one; tha...

the $k$-th semi total point graph $r^k(g)$ of a graph $g$, is a graph obtained from $g$ by adding $k$ vertices corresponding to each edge and connecting them to endpoint of edge considered. in this paper, we obtain formulae for the resistance distance and kirchhoff index of $r^k(g)$.

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...

Let G be a connected graph of order n . The Steiner distance d ( S ) set vertices is the minimum size subgraph that contains all For k ≤ , -diameter sdiam maximum among sets generalises classical diameter, which coincides with 2-diameter. problem determining given order, diameter and degree was first studied by Erdös Rényi. In this paper we consider corresponding for -diameter. Δ ∈ N define e a...

Minimum Steiner tree problem in a weighted graph is to find a least cost sub tree in the graph such that it contains special vertices called terminal points. Since this problem belongs to NP-Complete problems, many approximate algorithms including genetic algorithms and ant colony strategies have been designed for solving it. In this paper, we propose an algorithm based on distributed learning ...

Many models of genome rearrangement involve operations (e.g. inversions and translocations) that are self-inverse, and hence generate a group acting on the space of genomes. This gives a correspondence between genome arrangements and the elements of a group, and consequently, between evolutionary paths and walks on the Cayley graph. Many common methods for phylogeny reconstruction rely on calcu...

The problem of decomposing the block intersection graph of a Steiner triple system into triangles is considered. In the case when the block intersection graph has even degree, this is completely solved, while when the block intersection graph has odd degree, removal of some spanning subgraph of odd degree is necessary before the rest can be decomposed into triangles. In this case, some decompos...

Given a directed graph G = (V,E) and an integer k ≥ 1, a k-transitive-closure-spanner (k-TCspanner) of G is a directed graph H = (V,EH) that has (1) the same transitive-closure as G and (2) diameter at most k. In some applications, the shortcut paths added to the graph in order to obtain small diameter can use Steiner vertices, that is, vertices not in the original graph G. The resulting spanne...

A celebrated result of Thomassen states that not only can every planar graph be colored properly with five colors, but no matter how arbitrary palettes of five colors are assigned to vertices, one can choose a color from the corresponding palette for each vertex so that the resulting coloring is proper. This result is referred to as 5-choosability of planar graphs. Albertson asked whether Thoma...