Solving Vertex Cover in Polynomial Time on Hyperbolic Random Graphs

Vertex Cover Approximations on Random Graphs

The vertex cover problem is a classical NP-complete problem for which the best worst-case approximation ratio is 2 − o(1). In this paper, we use a collection of simple graph transformations, each of which guarantees an approximation ratio of 3 2 , to find approximate vertex covers for a large collection of randomly generated graphs. These reductions are extremely fast and even though they, by t...

ON THE EDGE COVER POLYNOMIAL OF CERTAIN GRAPHS

Let $G$ be a simple graph of order $n$ and size $m$.The edge covering of $G$ is a set of edges such that every vertex of $G$ is incident to at least one edge of the set. The edge cover polynomial of $G$ is the polynomial$E(G,x)=sum_{i=rho(G)}^{m} e(G,i) x^{i}$,where $e(G,i)$ is the number of edge coverings of $G$ of size $i$, and$rho(G)$ is the edge covering number of $G$. In this paper we stud...

Generating Random Hyperbolic Graphs in Subquadratic Time

Complex networks have become increasingly popular for modeling various real-world phenomena. Realistic generative network models are important in this context as they simplify complex network research regarding data sharing, reproducibility, and scalability studies. Random hyperbolic graphs are a very promising family of geometric graphs with unit-disk neighborhood in the hyperbolic plane. Prev...

On the Cover Time of Random Walks on Graphs

This article deals with random walks on arbitrary graphs. We consider the cover time of finite graphs. That is, we study the expected time needed for a random walk on a finite graph to visit every vertex at least once. We establish an upper bound of O(n 2) for the expectation of the cover time for regular (or nearly regular) graphs. We prove a lower bound of s log n) for the expected cover time...

The Cover Time of Random Regular Graphs