22 research outputs found
Polynomial Algorithm for Submap Isomorphism: Application to searching patterns in images
International audienceIn this paper, we address the problem of searching for a pattern in a plane graph, i.e., a planar drawing of a planar graph. To do that, we propose to model plane graphs with 2-dimensional combinatorial maps, which provide nice data structures for modelling the topology of a subdivision of a plane into nodes, edges and faces. We define submap isomorphism, we give a polynomial algorithm for this problem, and we show how this problem may be used to search for a pattern in a plane graph. First experimental results show the validity of this approach to efficiently search for patterns in images
Generalized Fitch Graphs II: Sets of Binary Relations that are explained by Edge-labeled Trees
Fitch graphs are digraphs that are explained by -edge-labeled rooted trees with leaf set : there is an arc if and only if the unique path in that connects the last common
ancestor of and with contains at least one edge
with label "1". In practice, Fitch graphs represent xenology relations, i.e.,
pairs of genes and for which a horizontal gene transfer happened along
the path from to .
In this contribution, we generalize the concept of Fitch graphs and consider
trees that are equipped with edge-labeling
that assigns to each edge a subset of colors. Given such a
tree, we can derive a map (or equivalently a set of
not necessarily disjoint binary relations), such that (or equivalently ) with , if and only if there is at least one edge with color from
to .
The central question considered here: Is a given map a Fitch
map, i.e., is there there an edge-labeled tree with
, and thus explains ?
Here, we provide a characterization of Fitch maps in terms of certain
neighborhoods and forbidden submaps. Further restrictions of Fitch maps are
considered. Moreover, we show that the least-resolved tree explaining a Fitch
map is unique (up to isomorphism). In addition, we provide a polynomial-time
algorithm to decide whether is a Fitch map and, in the
affirmative case, to construct the (up to isomorphism) unique least-resolved
tree that explains
The Kervaire-Laudenbach conjecture and presentations of simple groups
The statement ``no nonabelian simple group can be obtained from a nonsimple
group by adding one generator and one relator"
1) is equivalent to the Kervaire--Laudenbach conjecture;
2) becomes true under the additional assumption that the initial nonsimple
group is either finite or torsion-free.
Key words: Kervaire--Laudenbach conjecture, relative presentations, simple
groups, car motion, cocar comotion.
AMS MSC: 20E32, 20F05, 20F06.Comment: 20 pages, 13 figure