308,744 research outputs found
On perfect and unique maximum independent sets in graphs
summary:A perfect independent set of a graph is defined to be an independent set with the property that any vertex not in has at least two neighbors in . For a nonnegative integer , a subset of the vertex set of a graph is said to be -independent, if is independent and every independent subset of with is a subset of . A set of vertices of is a super -independent set of if is -independent in the graph , where is the bipartite graph obtained from by deleting all edges which are not incident with vertices of . It is easy to see that a set is -independent if and only if it is a maximum independent set and 1-independent if and only if it is a unique maximum independent set of . In this paper we mainly investigate connections between perfect independent sets and -independent as well as super -independent sets for and
Random Geometric Graphs and Isometries of Normed Spaces
Given a countable dense subset of a finite-dimensional normed space ,
and , we form a random graph on by joining, independently and with
probability , each pair of points at distance less than . We say that
is `Rado' if any two such random graphs are (almost surely) isomorphic.
Bonato and Janssen showed that in almost all are Rado. Our
main aim in this paper is to show that is the unique normed space
with this property: indeed, in every other space almost all sets are
non-Rado. We also determine which spaces admit some Rado set: this turns out to
be the spaces that have an direct summand. These results answer
questions of Bonato and Janssen.
A key role is played by the determination of which finite-dimensional normed
spaces have the property that every bijective step-isometry (meaning that the
integer part of distances is preserved) is in fact an isometry. This result may
be of independent interest
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