398 research outputs found
Reconstructing pedigrees: some identifiability questions for a recombination-mutation model
Pedigrees are directed acyclic graphs that represent ancestral relationships
between individuals in a population. Based on a schematic recombination
process, we describe two simple Markov models for sequences evolving on
pedigrees - Model R (recombinations without mutations) and Model RM
(recombinations with mutations). For these models, we ask an identifiability
question: is it possible to construct a pedigree from the joint probability
distribution of extant sequences? We present partial identifiability results
for general pedigrees: we show that when the crossover probabilities are
sufficiently small, certain spanning subgraph sequences can be counted from the
joint distribution of extant sequences. We demonstrate how pedigrees that
earlier seemed difficult to distinguish are distinguished by counting their
spanning subgraph sequences.Comment: 40 pages, 9 figure
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
Reconstructibility of matroid polytopes
We specify what is meant for a polytope to be reconstructible from its graph
or dual graph. And we introduce the problem of class reconstructibility, i.e.,
the face lattice of the polytope can be determined from the (dual) graph within
a given class. We provide examples of cubical polytopes that are not
reconstructible from their dual graphs. Furthermore, we show that matroid
(base) polytopes are not reconstructible from their graphs and not class
reconstructible from their dual graphs; our counterexamples include
hypersimplices. Additionally, we prove that matroid polytopes are class
reconstructible from their graphs, and we present a algorithm that
computes the vertices of a matroid polytope from its -vertex graph.
Moreover, our proof includes a characterisation of all matroids with isomorphic
basis exchange graphs.Comment: 22 pages, 5 figure
Maximal proper subgraphs of median graphs
AbstractFor a median graph G and a vertex v of G that is not a cut-vertex we show that G-v is a median graph precisely when v is not the center of a bipartite wheel, which is in turn equivalent with the existence of a certain edge elimination scheme for edges incident with v. This implies a characterization of vertex-critical (respectively, vertex-complete) median graphs, which are median graphs whose all vertex-deleted subgraphs are not median (respectively, are median). Moreover, two analogous characterizations for edge-deleted median graphs are given
Bounding Embeddings of VC Classes into Maximum Classes
One of the earliest conjectures in computational learning theory-the Sample
Compression conjecture-asserts that concept classes (equivalently set systems)
admit compression schemes of size linear in their VC dimension. To-date this
statement is known to be true for maximum classes---those that possess maximum
cardinality for their VC dimension. The most promising approach to positively
resolving the conjecture is by embedding general VC classes into maximum
classes without super-linear increase to their VC dimensions, as such
embeddings would extend the known compression schemes to all VC classes. We
show that maximum classes can be characterised by a local-connectivity property
of the graph obtained by viewing the class as a cubical complex. This geometric
characterisation of maximum VC classes is applied to prove a negative embedding
result which demonstrates VC-d classes that cannot be embedded in any maximum
class of VC dimension lower than 2d. On the other hand, we show that every VC-d
class C embeds in a VC-(d+D) maximum class where D is the deficiency of C,
i.e., the difference between the cardinalities of a maximum VC-d class and of
C. For VC-2 classes in binary n-cubes for 4 <= n <= 6, we give best possible
results on embedding into maximum classes. For some special classes of Boolean
functions, relationships with maximum classes are investigated. Finally we give
a general recursive procedure for embedding VC-d classes into VC-(d+k) maximum
classes for smallest k.Comment: 22 pages, 2 figure
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