6 research outputs found
Absolute reflexive retracts and absolute bipartite retracts
AbstractIt is a well-known phenomenon in the study of graph retractions that most results about absolute retracts in the class of bipartite (irreflexive) graphs have analogues about absolute retracts in the class of reflexive graphs, and vice versa. In this paper we make some observations that make the connection explicit. We develop four natural transformations between reflexive graphs and bipartite graphs which preserve the property of being an absolute retract, and allow us to derive results about absolute reflexive retracts from similar results about absolute bipartite retracts and conversely. Then we introduce generic notions that specialize to the appropriate concepts in both cases. This paves the way to a unified view of both theories, leading to absolute retracts of general (i.e., partially reflexive) graphs
Bi-complement Reducible Graphs
AbstractWe introduce a new family of bipartite graphs which is the bipartite analogue of the class ofcomplement reduciblegraphs orcographs. Abi-complement reduciblegraph orbi-cographis a bipartite graphG=(W∪B,E) that can be reduced to single vertices by recursively bi-complementing the edge set of all connected bipartite subgraphs. Thebi-complementedgraphḠbipofGis the graph having the same vertex setW∪BasG, while its edge set is equal toW×B−E. The aim of this paper is to show that there exists an equivalent definition of bi-cographs by three forbidden configurations. We also propose a tree representation for this class of graphs
Beyond Helly graphs: the diameter problem on absolute retracts
Characterizing the graph classes such that, on -vertex -edge graphs in
the class, we can compute the diameter faster than in time is an
important research problem both in theory and in practice. We here make a new
step in this direction, for some metrically defined graph classes.
Specifically, a subgraph of a graph is called a retract of if it is
the image of some idempotent endomorphism of . Two necessary conditions for
being a retract of is to have is an isometric and isochromatic
subgraph of . We say that is an absolute retract of some graph class
if it is a retract of any of which it is an
isochromatic and isometric subgraph. In this paper, we study the complexity of
computing the diameter within the absolute retracts of various hereditary graph
classes. First, we show how to compute the diameter within absolute retracts of
bipartite graphs in randomized time. For the
special case of chordal bipartite graphs, it can be improved to linear time,
and the algorithm even computes all the eccentricities. Then, we generalize
these results to the absolute retracts of -chromatic graphs, for every fixed
. Finally, we study the diameter problem within the absolute retracts
of planar graphs and split graphs, respectively