28 research outputs found
Characterizing graph classes by intersections of neighborhoods
The interplay between maxcliques (maximal cliques) and intersections of closed neighborhoods leads to new types of characterizations of several standard graph classes.
For instance, being hereditary clique-Helly is equivalent to every nontrivial maxclique containing the intersection of closed neighborhoods of two vertices of , and also to, in all induced subgraphs, every nontrivial maxclique containing a simplicial edge (an edge in a unique maxclique).
Similarly, being trivially perfect is equivalent to every maxclique containing the closed neighborhood of a vertex of ,
and also to, in all induced subgraphs, every maxclique containing a simplicial vertex.
Maxcliques can be generalized to maximal cographs, yielding a new characterization of ptolemaic graphs
Maxclique and unit disk characterizations of strongly chordal graphs
Maxcliques (maximal complete subgraphs) and unit disks (closed neighborhoods of vertices) sometime play almost interchangeable roles in graph theory. For instance, interchanging them makes two existing characterizations of chordal graphs into two new characterizations. More intriguingly, these characterizations of chordal graphs can be naturally strengthened to new characterizations of strongly chordal graphs.Facultad de Ciencias Exacta
Families of induced trees and their intersection graphs
This paper is inspired in the well known characterization of chordal graphs as the intersection graphs of subtrees of a tree. We consider families of induced trees of any graph and we prove that their recognition is NP-Complete. A consequence of this fact is that the concept of clique tree of chordal graphs cannot be widely generalized. Finally, we consider the fact that every graph is the intersection graph of induced trees of a bipartite graph and we characterize some classes that arise when we impose restrictions on the host bipartite graph.Facultad de Ciencias Exacta
The free splitting complex of a free group I: Hyperbolicity
We prove that the free splitting complex of a finite rank free group, also
known as Hatcher's sphere complex, is hyperbolic.Comment: 85 pages, including index and glossary. Many figures in the free
splitting complex, in the guise of commutative diagrams of maps between free
splittings. Changes from Version 1 to Version 2: Corrected an error in the
proof of Proposition 6.5, Step 2. Revamped the theory of free splitting units
in Section
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Polyhedra of small order and their Hamiltonian properties
We describe the results of an enumeration of several classes of polyhedra. The enumerated classes include polyhedra with up to 12 vertices and up to 26 edges, simplical polyhedra with up to 16 vertices, 4-connected polyhedra with up to 15 vertices, and bipartite polyhedra with up to 22 vertices.The results of the enumeration were used to systematically search for certain minimal non-Hamiltonian polyhedra. In particular, the smallest polyhedra satisfying certain toughness-like properties are presented here, as are the smallest non-Hamiltonian, 3-connected, Delaunay tessellations and triangulations. Improved upper and lower bounds on the size of the smallest non-Hamiltonian, inscribable polyhedra are also given
The rigidity of infinite graphs
A rigidity theory is developed for the Euclidean and non-Euclidean placements
of countably infinite simple graphs in R^d with respect to the classical l^p
norms, for d>1 and 1<p<\infty. Generalisations are obtained for the Laman and
Henneberg combinatorial characterisations of generic infinitesimal rigidity for
finite graphs in the Euclidean plane. Also Tay's multi-graph characterisation
of the rigidity of generic finite body-bar frameworks in d-dimensional
Euclidean space is generalised to the non-Euclidean l^p norms and to countably
infinite graphs. For all dimensions and norms it is shown that a generically
rigid countable simple graph is the direct limit of an inclusion tower of
finite graphs for which the inclusions satisfy a relative rigidity property.
For d>2 a countable graph which is rigid for generic placements in R^d may fail
the stronger property of sequential rigidity, while for d=2 the equivalence
with sequential rigidity is obtained from the generalised Laman
characterisations. Applications are given to the flexibility of non-Euclidean
convex polyhedra and to the infinitesimal and continuous rigidity of compact
infinitely-faceted simplicial polytopes.Comment: 51 page