26 research outputs found
On (2,3)-agreeable Box Societies
The notion of -agreeable society was introduced by Deborah Berg et
al.: a family of convex subsets of is called -agreeable if any
subfamily of size contains at least one non-empty -fold intersection. In
that paper, the -agreeability of a convex family was shown to imply the
existence of a subfamily of size with non-empty intersection, where
is the size of the original family and is an explicit
constant depending only on and . The quantity is called
the minimal \emph{agreement proportion} for a -agreeable family in
.
If we only assume that the sets are convex, simple examples show that
for -agreeable families in where . In this paper,
we introduce new techniques to find positive lower bounds when restricting our
attention to families of -boxes, i.e. cuboids with sides parallel to the
coordinates hyperplanes. We derive explicit formulas for the first non-trivial
case: the case of -agreeable families of -boxes with .Comment: 15 pages, 10 figure
A clique-difference encoding scheme for labelled k-path graphs
AbstractWe present in this paper a codeword for labelled k-path graphs. Structural properties of this codeword are investigated, leading to the solution of two important problems: determining the exact number of labelled k-path graphs with n vertices and locating a hamiltonian path in a given k-path graph in time O(n). The corresponding encoding scheme is also presented, providing linear-time algorithms for encoding and decoding
Algorithmic Aspects of a General Modular Decomposition Theory
A new general decomposition theory inspired from modular graph decomposition
is presented. This helps unifying modular decomposition on different
structures, including (but not restricted to) graphs. Moreover, even in the
case of graphs, the terminology ``module'' not only captures the classical
graph modules but also allows to handle 2-connected components, star-cutsets,
and other vertex subsets. The main result is that most of the nice algorithmic
tools developed for modular decomposition of graphs still apply efficiently on
our generalisation of modules. Besides, when an essential axiom is satisfied,
almost all the important properties can be retrieved. For this case, an
algorithm given by Ehrenfeucht, Gabow, McConnell and Sullivan 1994 is
generalised and yields a very efficient solution to the associated
decomposition problem
Lex-BFS and partition refinement, with applications to transitive orientation, interval graph recognition and consecutive ones testing
By making use of lexicographic breadth first search (Lex-BFS) and partition refinement with pivots, we obtain very simple algorithms for some well-known problems in graph theory. We give an O(n + m log n) algorithm for transitive orientation of a comparability graph, and simple linear algorithms to recognize interval graphs, convex graphs, Y -semichordal graphs and matrices that have the consecutive-ones property. Previous approaches to these problems used difficult preprocessing steps, such as computing PQ trees or modular decomposition. The algorithms we give are easy to understand and straightforward to prove. They do not make use of sophisticated data structures, and the complexity analysis is straightforward. Keywords algorithm, data-structure, partition refinement, graph, boolean matrix 1 Introduction Some efficient algorithms for various classes of graphs and boolean matrices are presented. These classes are comparability, chordal, interval graphs and their complements. To th..