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Minimal Universal Bipartite Graphs
Vadim V. Lozin, Gábor Rudolf RUTCOR – Rutgers University
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Hereditary classes
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Universal Graphs Trivial bounds:
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Optimal universal graphs
A sequence of universal X-graphs is asimptotically optimal if optimal in order if
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Index of a hereditary class
V.E. Alekseev (1992):
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Unitary classes Graph classes of the same index constitute a stratum
Classes of index 1: unitary stratum Contains many interesting classes: Forests Planar graphs Interval graphs Permutation graphs Line graphs Etc.
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Isometry of graph classes
The equivalence classes are called layers Scheinarman and Zito (1994): The first four layers in the unitary stratum are:
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Exponential classes Alexeev (1997) For each exponential class X there is a constant p such that every graph G in X can be partitioned into at most p subsets each of which is either an independent set or a clique and between any two subsets there are either all possible edges or none of them. Asimptotically optimal universal graphs exist
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Minimal factorial classes of bipartite graphs
Alexeev (1997) The three minimal factorial classes of bipartite graphs are: Asimptotically optimal universal graphs exist for all these classes (trivial for ).
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Canonical graphs
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Universal chain graphs
is an n-universal chain graph
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Universal chain graphs
is an n-universal chain graph
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Universal chain graphs
is an n-universal chain graph
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Universal biparite permutation graphs
Brandtstädt, Lozin (2003): is an n-universal bipartite permutation graph; it is optimal in order
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Unit interval graphs
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General bipartite graphs
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Let us first consider :
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Optimality
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