The Hosoya polynomial of double weighted graphs

Abstract

The modified Hosoya polynomial of double weighted graphs, i.e. edge and vertex weighted graphs, is introduced that enables derivation of closed expressions for Hosoya polynomial of some special graphs including unicyclic graphs. Furthermore, the Hosoya polynomial is given as a sum of edge contributions generalizing well known analogous results for the Wiener number. A linear algorithm for computing the Hosoya polynomial on cactus graphs is provided. Hosoya polynomial is extensively studied in chemical graph theory, and in particular its weighted versions have interesting applications in theory of communication networks.

A Linear Time Algorithm for Weighted k-Fair Domination Problem in Cactus Graphs

“…In this paper, the first linear time algorithm for computing the k-fair domination number of a cactus graph is designed. While the basic ideas follow the already known procedures elaborated already in [11][12][13], it should be emphasized that some novel observations are used that result in a simplified and faster algorithm. These novel observations may also be used to improve the efficiency of algorithms for related problems developed in [11][12][13].…”

“…Following the idea of the algorithms in [11][12][13], at the beginning of the algorithm we have to set the initial values of all parameters wfd + k and wfd k for = k, k − 1, … , 1, 0 at every vertex of a given cactus graph G as a vertex is a singleton graph (a graph with one vertex and no edges). For every v ∈ V(G) the initialization is…”

“…Recent examples include the weighted domination problem and the weighted independent k-domination problem, which can be solved in linear time for cactus graphs (see [11] and [12], respectively). Further examples of linear algorithms on cacti include computation of the Hosoya polynomial [13] and determination of the broadcast time for all vertices in k-restricted cactus graph [8]. The linear algorithms for the weighted domination problem, for the weighted independent k-domination problem, and for the Hosoya polynomial share some common ideas.…”

“…Note that listing all such sets can in general clearly not be done in polynomial time. While the basic ideas follow the above mentioned [11][12][13], it should be emphasized that some novel observations are used that result in a simplified and faster algorithm. We only wish to mention in passing that these results also may be used to improve efficiency of [11][12][13].…”

“…While the basic ideas follow the above mentioned [11][12][13], it should be emphasized that some novel observations are used that result in a simplified and faster algorithm. We only wish to mention in passing that these results also may be used to improve efficiency of [11][12][13]. As already mentioned, we believe that the results here can be used to solve the open question about complexity of the fair domination problem.…”