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References etc.

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Antti H S Laaksonen 2017-02-25 19:39:04 +02:00
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6
chapter20.tex
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@ -930,7 +930,7 @@ The maximum flow of this graph is as follows:
\index{Hall's theorem} \index{Hall's theorem}
\index{perfect matching} \index{perfect matching}
\key{Hall's theorem} can be used to find out \key{Hall's theorem} \cite{hal35} can be used to find out
whether a bipartite graph has a matching whether a bipartite graph has a matching
that contains all left or right nodes. that contains all left or right nodes.
If the number of left and right nodes is the same, If the number of left and right nodes is the same,
@ -1020,7 +1020,7 @@ has at least one endpoint in the set.
In a general graph, finding a minimum node cover In a general graph, finding a minimum node cover
is a NP-hard problem. is a NP-hard problem.
However, if the graph is bipartite, However, if the graph is bipartite,
\key{Kőnig's theorem} tells us that \key{Kőnig's theorem} \cite{kon31} tells us that
the size of a minimum node cover the size of a minimum node cover
and the size of a maximum matching are always equal. and the size of a maximum matching are always equal.
Thus, we can calculate the size of a minimum node cover Thus, we can calculate the size of a minimum node cover
@ -1409,7 +1409,7 @@ An \key{antichain} is a set of nodes of a graph
such that there is no path such that there is no path
from any node to another node from any node to another node
using the edges of the graph. using the edges of the graph.
\key{Dilworth's theorem} states that \key{Dilworth's theorem} \cite{dil50} states that
in a directed acyclic graph, the size of in a directed acyclic graph, the size of
a minimum general path cover a minimum general path cover
equals the size of a maximum antichain. equals the size of a maximum antichain.

17
list.tex
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@ -44,6 +44,11 @@
A note on two problems in connexion with graphs. A note on two problems in connexion with graphs.
\emph{Numerische Mathematik}, 1(1):269--271, 1959. \emph{Numerische Mathematik}, 1(1):269--271, 1959.
\bibitem{dil50}
R. P. Dilworth.
A decomposition theorem for partially ordered sets.
\emph{Annals of Mathematics}, 51(1):161--166, 1950.
\bibitem{dir52} \bibitem{dir52}
G. A. Dirac. G. A. Dirac.
Some theorems on abstract graphs. Some theorems on abstract graphs.
@ -117,6 +122,13 @@
Computer Science and Computational Biology}, Computer Science and Computational Biology},
Cambridge University Press, 1997. Cambridge University Press, 1997.
\bibitem{hal35}
P. Hall.
On representatives of subsets.
\emph{Journal London Mathematical Society} 10(1):26--30, 1935.
On representatives of subsets. J. London Math. Soc, 10(1), 26-30.
\bibitem{hel62} \bibitem{hel62}
M. Held and R. M. Karp. M. Held and R. M. Karp.
A dynamic programming approach to sequencing problems. A dynamic programming approach to sequencing problems.
@ -164,6 +176,11 @@
D. E. Knuth. D. E. Knuth.
\emph{The Art of Computer Programming. Volume 3: Sorting and Searching}, Addison–Wesley, 1998 (2nd edition). \emph{The Art of Computer Programming. Volume 3: Sorting and Searching}, Addison–Wesley, 1998 (2nd edition).
\bibitem{kon31}
D. Kőnig.
Gráfok és mátrixok.
\emph{Matematikai és Fizikai Lapok}, 38(1):116--119, 1931.
\bibitem{kru56} \bibitem{kru56}
J. B. Kruskal. J. B. Kruskal.
On the shortest spanning subtree of a graph and the traveling salesman problem. On the shortest spanning subtree of a graph and the traveling salesman problem.