Formulas for the Wiener number and the Hosoya-Wiener polynomial of edge and vertex weighted graphs are given in terms of edge and path contributions. For a rooted tree, the Hosoya-Wiener polynomial is expressed as a sum of vertex contributions. Finally, a recursive formula for computing the Hosoya-Wiener polynomial of a weighted tree is given.

In this paper, we study the computational complexity and approximation complexity of the connected set-cover problem. We derive necessary and sufficient conditions for the connected set-cover problem to have a polynomial-time algorithm. We also present a sufficient condition for the existence of a (1 + ln δ)-approximation. In addition, one such (1 + ln δ)-approximation algorithm for this proble...

In this paper we fix 7 types of undirected graphs: paths, paths with prescribed endvertices, circuits, forests, spanning trees, (not necessarily spanning) trees and cuts. Given an undirected graph G = (V, E) and two “object types” A and B chosen from the alternatives above, we consider the following questions. Packing problem: can we find an object of type A and one of type B in the edge set E ...

Partial constraint satisfaction [5] was widely studied in the 90’s, and notably Max-CSP solving algorithms [21, 20, 1, 10]. These algorithms compute a lower bound of violated constraints without using propagation. Therefore, recent methods focus on the exploitation of propagation mechanisms to improve the solving process. Soft arc-consistency algorithms [11, 18, 19] propagate inconsistency coun...

We consider the parameterized problem of counting all matchings with exactly k edges in a given input graph G. This problem is #W[1]-hard (Curticapean, ICALP 2013), so it is unlikely to admit f(k) · nO(1) time algorithms. We show that #W[1]-hardness persists even when the input graph G comes from restricted graph classes, such as line graphs and bipartite graphs of arbitrary constant girth and ...

We consider the Minimum Vertex Cover problem in intersection graphs of axis-parallel rectangles on the plane. We present two algorithms: The first is an EPTAS for non-crossing rectangle families, rectangle families R where R1 \ R2 is connected for every pair of rectangles R1, R2 ∈ R. This algorithm extends to intersection graphs of pseudo-disks. The second algorithm achieves a factor of (1.5 + ...

Given a graph G and an integer k, the paw-free completion problem asks whether it is possible to add at most k edges make paw-free. The edge deletion defined analogously. Sandeep Sivadasan (IPEC 2015) asked these problems admit polynomial kernels. We answer both questions affirmatively by presenting, respectively, O(k)-vertex O(k4)-vertex kernels for them. This part of ongoing program that aims...

We consider a strategic game with two classes of confronting randomized players on a graph G(V,E): ν attackers, each choosing vertices and wishing to minimize the probability of being caught, and a defender, who chooses edges and gains the expected number of attackers it catches. The Price of Defense is the worst-case ratio, over all Nash equilibria, of the optimal gain of the defender over its...