50 research outputs found
Modular Decomposition and the Reconstruction Conjecture
We prove that a large family of graphs which are decomposable with respect to
the modular decomposition can be reconstructed from their collection of
vertex-deleted subgraphs.Comment: 9 pages, 2 figure
Inversion dans les tournois
We consider the transformation reversing all arcs of a subset of the
vertex set of a tournament . The \emph{index} of , denoted by , is
the smallest number of subsets that must be reversed to make acyclic. It
turns out that critical tournaments and -critical tournaments can be
defined in terms of inversions (at most two for the former, at most four for
the latter). We interpret as the minimum distance of to the
transitive tournaments on the same vertex set, and we interpret the distance
between two tournaments and as the \emph{Boolean dimension} of a
graph, namely the Boolean sum of and . On vertices, the maximum
distance is at most , whereas , the maximum of over the
tournaments on vertices, satisfies , for . Let (resp.
) be the class of finite (resp. at most
countable) tournaments such that . The class is determined by finitely many obstructions. We give a
morphological description of the members of and a
description of the critical obstructions. We give an explicit description of an
universal tournament of the class .Comment: 6 page
Graphs whose indecomposability graph is 2-covered
Given a graph , a subset of is an interval of provided
that for any and , if and only
if . For example, , and are
intervals of , called trivial intervals. A graph whose intervals are trivial
is indecomposable; otherwise, it is decomposable. According to Ille, the
indecomposability graph of an undirected indecomposable graph is the graph
whose vertices are those of and edges are the unordered
pairs of distinct vertices such that the induced subgraph is indecomposable. We characterize the indecomposable
graphs whose admits a vertex cover of size 2.Comment: 31 pages, 5 figure