-
In 1735, Euler presented a
solution to the problem known as the Seven Bridges
of K?nigsberg.[35] The city of
K?nigsberg,
Prussia was set
on
the Pregel
River,
and
included
two
large
islands
that
were
connected
to
each
other and the mainland by seven
bridges. The problem is to decide whether it is
possible to follow a path that
crosses
each bridge exactly once and returns to the
starting point. It is not possible: there is no
Eulerian circuit.
This solution is
considered to be the first theorem of graph
theory, specifically of planar graph theory.[35]
Euler also discovered the formula V ? E
+ F = 2 relating the number of vertices, edges and
faces of a conv
ex
polyhedron,[36]
and
hence
of
a
planar
graph.
The
constant
in
this
formula
is
now
known
as
the
Euler
characteristic for the
graph (or other mathematical object), and is
related to the genus of the object.[37] The
study and generalization of this
formula, specifically by Cauchy[38] and
L'Huillier,[39] is at the origin of topology.
ickson
Frederickson proposed
a heuristics that solve rural postman. The
algorithm mainly consists of two steps: it firstly
find the shortest tree over connected
components, and then match odd-degree vertices.
He is known for
characterizing the degree sequences of undirected
graphs,[3] for formulating the Steiner tree
problem on networks, and for his work
on facility location problems on networks.
There always exists an optimal solution
located at vertices.
lzer
Hierholzer proved that a graph has an
Eulerian cycle if and only if it is connected and
every vertex has an even
degree. This
result had been given, without proof, by Leonhard
Euler in 1736. He also proposed a algorithm for
how to find the Eulerian.
l
In computer science, his best known
work is Kruskal's algorithm for computing the
minimal spanning tree (MST)
of a
weighted graph. The algorithm first orders the
edges by weight and then proceeds through the
ordered list
adding an edge to the
partial MST provided that adding the new edge does
not create a cycle. Minimal spanning
trees
have
applications
to
the
construction
and
pricing
of
communication
networks.
In
combinatorics,
he
is
known for
Kruskal's tree theorem (1960), which is also
interesting from a mathematical logic perspective
since it
can only be proved
nonconstructively.
Kruskal's algorithm
is a greedy algorithm in graph theory that finds a
minimum spanning tree for a connected
weighted graph. This means it finds a
subset of the edges that forms a tree that
includes every vertex, where the
total
weight of all the edges in the tree is minimized.
If the graph is not connected, then it finds a
minimum
spanning forest (a minimum
spanning tree for each connected component).
E. Weiszfeld
Weiszfeld's
algorithm after the work of Endre Weiszfeld,[4] is
a form of iteratively re-weighted least squares.
This
algorithm
defines
a
set
of
weights
that
are
inversely
proportional
to
the
distances
from
the
current
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