An asteroidal triple (AT) is a set of three vertices such that there is a path between any pair of them avoiding the closed neighborhood of the third. A graph is called AT-free if it does not have an AT. We show that there is an O(n4) time algorithm to compute the maximum weight of an independent set for AT-free graphs. Furthermore, we obtain O(n4) time algorithms to solve the independent domin...

We integrated the orbital evolution of 30,000 Jupiter-family comets, 1300 resonant asteroids, and 7000 asteroidal, trans-Neptunian, and cometary dust particles. For initial orbital elements of bodies close to those of Comets 2P, 10P, 44P, and 113P, a few objects got Earthcrossing orbits with semi-major axes a<2 AU and moved in such orbits for more than 1 Myr (up to tens or even hundreds of Myrs...

The orbital evolution of Jupiter-family comets (JFCs), resonant asteroids, and asteroidal, trans-Neptunian, and cometary dust particles under the gravitational influence of planets was integrated. For dust particles we also considered radiation pressure, Poynting-Robertson drag, and solar wind drag. The probability of a collision of one former JFC with a terrestrial planet can be greater than a...

An asteroidal triple is a set of three independent vertices such that between any two of them there exists a path that avoids the neighl.)ourhood of the third. Graphs that do not. co~,tain an asteroidal triple are called asteroidal triple-free (AT-free) graphs. AT-free graphs strictly contain the well-known class of cocomparability graphs, and are not necessarily perfect.. We present efficient,...

An axis parallel d-dimensional box is the Cartesian product R1×R2×· · ·×Rd where each Ri is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer d such that G can be represented as the intersection graph of a collection of d-dimensional boxes. An axis parallel unit cube in d-dimensional space or a d-cube is defined as the Cartesian product R1 ...

An asteroidal triple is a stable set of three vertices such that each pair is connected by a path avoiding the neighborhood of the third vertex. Asteroidal triples play a central role in a classical characterization of interval graphs by Lekkerkerker and Boland. Their result says that a chordal graph is an interval graph if and only if it does not contain an asteroidal triple. In this paper, we...