The Minimum Wiener Connector

The Wiener index of a graph is the sum of all pairwise shortest-path distances between its vertices. In this paper we study the novel problem of finding a minimum Wiener connector: given a connected graph $G=(V,E)$ and a set $Q\subseteq V$ of query vertices, find a subgraph of $G$ that connects all query vertices and has minimum Wiener index. We show that The Minimum Wiener Connector admits a polynomial-time (albeit impractical) exact algorithm for the special case where the number of query vertices is bounded. We show that in general the problem is NP-hard, and has no PTAS unless $\mathbf{P} = \mathbf{NP}$. Our main contribution is a constant-factor approximation algorithm running in time $\widetilde{O}(|Q||E|)$. A thorough experimentation on a large variety of real-world graphs confirms that our method returns smaller and denser solutions than other methods, and does so by adding to the query set $Q$ a small number of important vertices (i.e., vertices with high centrality).Comment: Published in Proceedings of the 2015 ACM SIGMOD International Conference on Management of Dat

Extremal Values of Ratios: Distance Problems vs. Subtree Problems in Trees

The authors discovered a dual behaviour of two tree indices, the Wiener index and the number of subtrees, for a number of extremal problems [Discrete Appl. Math. 155 (3) 2006, 374-385; Adv. Appl. Math. 34 (2005), 138-155]. Barefoot, Entringer and Székely [Discrete Appl. Math. 80 (1997), 37-56] determined extremal values of σT(w)/σT(u), σT(w)/σT(v), σ(T)/σT(v), and σ(T)/σT(w), where T is a tree on n vertices, v is in the centroid of the tree T, and u,w are leaves in T. In this paper we test how far the negative correlation between distances and subtrees go if we look for the extremal values of FT(w)/FT(u), FT(w)/FT(v), F(T)/FT(v), and F(T)/FT(w), where T is a tree on n vertices, v is in the subtree core of the tree T, and u,w are leaves in T-the complete analogue of [Discrete Appl. Math. 80 (1997), 37-56], changing distances to the number of subtrees. We include a number of open problems, shifting the interest towards the number of subtrees in graphs

Functions on Adjacent Vertex Degrees of Trees with Given Degree Sequence

In this note we consider a discrete symmetric function f(x, y) where f(x; a) + f(y, b) ≥ f(y, a) + f(x, b) for any x ≥ y and a ≥ b, associated with the degrees of adjacent vertices in a tree. The extremal trees with respect to the corresponding graph invariant, defined as Σ uv∈E(T) f(deg(u), deg(v)), are characterized by the “greedy tree” and “alternating greedy tree”. This is achieved through simple generalizations of previously used ideas on similar questions. As special cases, the already known extremal structures of the Randić index follow as corollaries. The extremal structures for the relatively new sum-connectivity index and harmonic index also follow immediately, some of these extremal structures have not been identified in previous studies