A distance-hereditary graph is a connected graph in which every induced path is isometric, i.e., the distance of any two vertices in an induced path equals their distance in the graph. We present a linear time labeling algorithm for the minimum cardinality connected r-dominating set and Steiner tree problems on distance-hereditary graphs. q 1998 John Wiley & Sons, Inc. Networks 31: 177–182, 1998

The problem of constructing an optimal Steiner tree is NP-complete. Mapping the problem to standard Cartesian coordinates, the cost of an optimal Steiner tree is defined as the minimal rectilinear distance of all the edges. The heuristic proposed in this paper then finds a close approximation in fast (polynomial) time. The basis of exploits the idea of triangular distance, that is, using distan...

Distance-hereditary graphs are graphs in which every two vertices have the same distance in every connected induced subgraph containing them. This paper studies distance-hereditary graphs from an algorithmic viewpoint. In particular, we present linear-time algorithms for finding a minimum weighted connected dominating set and a minimum vertex-weighted Steiner tree in a distance-hereditary graph...

A.D. Forbes, M.J. Grannell and T.S. Griggs Department of Mathematics The Open University Walton Hall, Milton Keynes MK7 6AA UNITED KINGDOM [email protected] [email protected] [email protected] Abstract In [8], Quattrochi and Rinaldi introduced the idea of n−1 isomorphism between Steiner systems. In this paper we study this concept in the context of Steiner triple systems. Th...

Let G be a connected graph and S ⊆ V (G). Then the Steiner distance of S, denoted by dG(S), is the smallest number of edges in a connected subgraph of G containing S. Such a subgraph is necessarily a tree called a Steiner tree for S. The Steiner interval for a set S of vertices in a graph, denoted by I (S) is the union of all vertices that belong to some Steiner tree for S. If S = {u, v}, then ...

The eccentric sequence of a connected graph \(G\) is the nondecreasing eccentricities its vertices. Wiener index sum distances between all unordered pairs vertices \(G\). unique trees that minimise among with given were recently determined by present authors. In this paper we show these results hold not only for index, but large class distance-based topological indices which term Wiener-type in...

Let G be a graph. The Steiner distance of $$W\subseteq V(G)$$ is the minimum size connected subgraph containing W. Such necessarily tree called W-tree. set $$A\subseteq k-Steiner general position if $$V(T_B)\cap A = B$$ holds for every $$B\subseteq A$$ cardinality k, and B-tree $$T_B$$ . number $$\mathrm{sgp}_k(G)$$ largest in G. cliques are introduced used to bound from below. determined trees...

The smallest tree that contains all vertices of a subset W of V (G) is called a Steiner tree. The number of edges of such a tree is the Steiner distance of W and union of all Steiner trees of W form a Steiner interval. Both of them are described for the lexicographic product in the present work. We also give a complete answer for the following invariants with respect to the Steiner convexity: t...