208,847 research outputs found
On the One dimensional Poisson Random Geometric Graph
Given a Poisson process on a bounded interval, its random geometric graph is
the graph whose vertices are the points of the Poisson process and edges exist
between two points if and only if their distance is less than a fixed given
threshold. We compute explicitly the distribution of the number of connected
components of this graph. The proof relies on inverting some Laplace
transforms
Components of domino tilings under flips in quadriculated cylinder and torus
In a region consisting of unit squares, a domino is the union of two
adjacent squares and a (domino) tiling is a collection of dominoes with
disjoint interior whose union is the region. The flip graph is
defined on the set of all tilings of such that two tilings are adjacent if
we change one to another by a flip (a rotation of a pair of
side-by-side dominoes). It is well-known that is connected
when is simply connected. By using graph theoretical approach, we show that
the flip graph of quadriculated cylinder is still connected,
but the flip graph of quadriculated torus is disconnected and
consists of exactly two isomorphic components. For a tiling , we associate
an integer , forcing number, as the minimum number of dominoes in
that is contained in no other tilings. As an application, we obtain that the
forcing numbers of all tilings in quadriculated cylinder and
torus form respectively an integer interval whose maximum value is
Induced Subgraph Isomorphism on proper interval and bipartite permutation graphs *
Abstract Given two graphs G and H as input, the Induced Subgraph Isomorphism (ISI) problem is to decide whether G has an induced subgraph that is isomorphic to H. This problem is NP-complete already when G and H are restricted to disjoint unions of paths, and consequently also NP-complete on proper interval graphs and on bipartite permutation graphs. We show that ISI can be solved in polynomial time on proper interval graphs and on bipartite permutation graphs, provided that H is connected. As a consequence, we obtain that ISI is fixed-parameter tractable on these two graph classes, when parametrised by the number of connected components of H. Our results contrast and complement the following known results: W [1]-hardness of ISI on interval graphs when parametrised by the number of vertices of H, NP-completeness of ISI on connected interval graphs and on connected permutation graphs, and NP-completeness of Subgraph Isomorphism on connected proper interval graphs and connected bipartite permutation graphs
- …