Given a Steiner triple system S, we say that a cubic graph G is S-colourable if its edges can be coloured by points of S in such way that the colours of any three edges meeting at a vertex form a triple of S. We prove that there is Steiner triple system U of order 21 which is universal in the sense that every simple cubic graph is U-colourable. This improves the result of Grannell et al. [J. Gr...

Let G be a connected simple (molecular) graph. The distance d(u, v) between two vertices u and v of G is equal to the length of a shortest path that connects u and v. In this paper we compute some distance based topological indices of H-Phenylenic nanotorus. At first we obtain an exact formula for the Wiener index. As application we calculate the Schultz index and modified Schultz index of this...

We tackle the problem of constructing increasing-chord graphs spanning point sets. We prove that, for every point set P with n points, there exists an increasing-chord planar graph with O(n) Steiner points spanning P . Further, we prove that, for every convex point set P with n points, there exists an increasingchord graph with O(n log n) edges (and with no Steiner points) spanning P .

We study the approximability of three versions of the Steiner tree problem. For the rst one where the input graph is only supposed connected, we show that it is not approximable within better than |V \N |−2 for any 2 ∈ (0, 1), where V and N are the vertex-set of the input graph and the set of terminal vertices, respectively. For the second of the Steiner tree versions considered, the one where ...

‎The textit{metric dimension} of a connected graph $G$ is the minimum number of vertices in a subset $B$ of $G$ such that all other vertices are uniquely determined by their distances to the vertices in $B$‎. ‎In this case‎, ‎$B$ is called a textit{metric basis} for $G$‎. ‎The textit{basic distance} of a metric two dimensional graph $G$ is the distance between the elements of $B$‎. ‎Givi...

Routing is one of the important steps in very/ultra large-scale integration (VLSI/ULSI) physical design. Rectilinear Steiner minimal tree (RSMT) construction is an essential part of routing. Macro cells, IP blocks, and pre-routed nets are often regarded as obstacles in the routing phase. Obstacle-avoiding RSMT (OARSMT) algorithms are useful for practical routing applications. However, OARSMT al...

We study the approximability of three versions of the Steiner tree problem. For the rst one where the input graph is only supposed connected, we show that it is not approximable within better than |V \N |−ǫ for any ǫ ∈ (0, 1), where V and N are the vertex-set of the input graph and the set of terminal vertices, respectively. For the second of the Steiner tree versions considered, the one where ...

For a metric graph G = (V,E) and R ⊂ V , the internal Steiner minimum tree problem asks for a minimum weight Steiner tree spanning R such that every vertex in R is not a leaf. This note shows a simple polynomial-time 2ρapproximation algorithm, in which ρ is the approximation ratio for the Steiner minimum tree problem. The result improves the approximation ratio 2ρ+ 1 in [3].

We prove a limit theorem for the total Steiner k-distance of a random b-ary recursive tree with weighted edges. The total Steiner k-distance is the sum of all Steiner k-distances in a tree and it generalises the Wiener index. The limit theorem is obtained by using a limit theorem in the general setting of the contraction method. In order to use the contraction method we prove a recursion formul...

We consider the vertex-weighted version of the undirected Steiner tree problem. In this problem, a cost is incurred both for the vertices and the edges present in the Steiner tree. We completely describe the associated polytope by linear inequalities when the underlying graph is series-parallel. For general graphs, this formulation can be interpreted as a (partial) extended formulation for the ...