Given a graph G = (V; E) and a positive integer k, the Phylogenetic k-Root Problem asks for a (unrooted) tree T without degree-2 nodes such that its leaves are labeled by V and (u; v) 2 E if and only if dT (u; v) k. If the vertices in V are also allowed to be internal nodes in T , then we have the Steiner k-Root Problem. Moreover, if a particular subset S of V are required to be internal nodes ...

Given a weighted graph G = (V,E) and a subset R of V , a Steiner tree in G is a tree which spans all vertices in R. The vertices in V \R are called Steiner vertices. A full Steiner tree is a Steiner tree in which each vertex of R is a leaf. The full Steiner tree problem is to find a full Steiner tree with minimum weight. The bottleneck full Steiner tree problem is to find a full Steiner tree wh...

We investigate a practical variant of the well-known graph Steiner tree problem. In this variant, every target vertex is required to be a leaf vertex in the solution Steiner tree. We present hardness results for this variant as well as a polynomial time approximation algorithm with performance ratio ρ + 2, where ρ is the best-known approximation ratio for the graph Steiner tree problem.  2002 ...

a set $s$ of vertices in a graph $g=(v,e)$ is called a total$k$-distance dominating set if every vertex in $v$ is withindistance $k$ of a vertex in $s$. a graph $g$ is total $k$-distancedomination-critical if $gamma_{t}^{k} (g - x) < gamma_{t}^{k}(g)$ for any vertex $xin v(g)$. in this paper,we investigate some results on total $k$-distance domination-critical of graphs.

let g be a simple connected graph and {v1, v2, …, vk} be the set of pendent (vertices ofdegree one) vertices of g. the reduced distance matrix of g is a square matrix whose (i,j)–entry is the topological distance between vi and vj of g. in this paper, we obtain the spectrumof the reduced distance matrix of thorn graph of g, a graph which obtained by attaching somenew vertices to pendent vertice...

In this paper we introduce the notion of a bicolour graph of a Steiner triple system. Given an arbitrary colouring of the points of a Steiner triple system, the bicolour graph is the graph obtained on the point set of the system by placing an edge between two points if they have the same colour and appear in a triple with a point of a di4erent colour. The case where the graph is Kv was partiall...

We study the Steiner Tree problem, in which a set of terminal vertices needs to be connected in the cheapest possible way in an edge-weighted graph. This problem has been extensively studied from the viewpoint of approximation and also parametrization. In particular, on one hand Steiner Tree is known to be APX-hard, and W[2]-hard on the other, if parameterized by the number of non-terminals (St...