70 research outputs found
On the swap-distances of different realizations of a graphical degree sequence
One of the first graph theoretical problems which got serious attention
(already in the fifties of the last century) was to decide whether a given
integer sequence is equal to the degree sequence of a simple graph (or it is
{\em graphical} for short). One method to solve this problem is the greedy
algorithm of Havel and Hakimi, which is based on the {\em swap} operation.
Another, closely related question is to find a sequence of swap operations to
transform one graphical realization into another one of the same degree
sequence. This latter problem got particular emphases in connection of fast
mixing Markov chain approaches to sample uniformly all possible realizations of
a given degree sequence. (This becomes a matter of interest in connection of --
among others -- the study of large social networks.) Earlier there were only
crude upper bounds on the shortest possible length of such swap sequences
between two realizations. In this paper we develop formulae (Gallai-type
identities) for these {\em swap-distance}s of any two realizations of simple
undirected or directed degree sequences. These identities improves considerably
the known upper bounds on the swap-distances.Comment: to be publishe
A simple Havel-Hakimi type algorithm to realize graphical degree sequences of directed graphs
One of the simplest ways to decide whether a given finite sequence of
positive integers can arise as the degree sequence of a simple graph is the
greedy algorithm of Havel and Hakimi. This note extends their approach to
directed graphs. It also studies cases of some simple forbidden edge-sets.
Finally, it proves a result which is useful to design an MCMC algorithm to find
random realizations of prescribed directed degree sequences.Comment: 11 pages, 1 figure submitted to "The Electronic Journal of
Combinatorics
Caterpillar dualities and regular languages
We characterize obstruction sets in caterpillar dualities in terms of regular
languages, and give a construction of the dual of a regular family of
caterpillars. We show that these duals correspond to the constraint
satisfaction problems definable by a monadic linear Datalog program with at
most one EDB per rule
A finite word poset : In honor of Aviezri Fraenkel on the occasion of his 70th birthday
Our word posets have �nite words of bounded length as their elements, with
the words composed from a �nite alphabet. Their partial ordering follows from the
inclusion of a word as a subsequence of another word. The elemental combinatorial
properties of such posets are established. Their automorphism groups are determined
(along with similar result for the word poset studied by Burosch, Frank and
R¨ohl [4]) and a BLYM inequality is veri�ed (via the normalized matching property)
On the average rank of LYM-sets
Let S be a finite set with some rank function r such that the Whitney numbers wi = |{x S|r(x) = i}| are log-concave. Given so that wk − 1 < wk wk + m, set W = wk + wk + 1 + … + wk + m. Generalizing a theorem of Kleitman and Milner, we prove that every F S with cardinality |F| W has average rank at least kwk + … + (k + m) wk + m/W, provided the normalized profile vector x1, …, xn of F satisfies the following LYM-type inequality: x0 + x1 + … + xn m + 1
AZ-identities and Strict 2-part Sperner Properties of Product Posets
One of the central issues in extremal set theory is Sperner's theorem and its
generalizations. Among such generalizations is the best-known BLYM inequality
and the Ahlswede--Zhang (AZ) identity which surprisingly generalizes the BLYM
inequality into an identity. Sperner's theorem and the BLYM inequality has been
also generalized to a wide class of posets. Another direction in this research
was the study of more part Sperner systems. In this paper we derive AZ type
identities for regular posets. We also characterize all maximum 2-part Sperner
systems for a wide class of product posets
Balanced Vertices in Trees and a Simpler Algorithm to Compute the Genomic Distance
This paper provides a short and transparent solution for the covering cost of
white-grey trees which play a crucial role in the algorithm of Bergeron {\it et
al.}\ to compute the rearrangement distance between two multichromosomal
genomes in linear time ({\it Theor. Comput. Sci.}, 410:5300-5316, 2009). In the
process it introduces a new {\em center} notion for trees, which seems to be
interesting on its own.Comment: 6 pages, submitte
- …