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Minimal Universal Bipartite Graphs

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Presentation on theme: "Minimal Universal Bipartite Graphs"— Presentation transcript:

1 Minimal Universal Bipartite Graphs
Vadim V. Lozin, Gábor Rudolf RUTCOR – Rutgers University

2 Hereditary classes

3 Universal Graphs Trivial bounds:

4 Optimal universal graphs
A sequence of universal X-graphs is asimptotically optimal if optimal in order if

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6 Index of a hereditary class
V.E. Alekseev (1992):

7 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.

8 Isometry of graph classes
The equivalence classes are called layers Scheinarman and Zito (1994): The first four layers in the unitary stratum are:

9 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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12 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 ).

13 Canonical graphs

14 Universal chain graphs
is an n-universal chain graph

15 Universal chain graphs
is an n-universal chain graph

16 Universal chain graphs
is an n-universal chain graph

17 Universal biparite permutation graphs
Brandtstädt, Lozin (2003): is an n-universal bipartite permutation graph; it is optimal in order

18 Unit interval graphs

19 General bipartite graphs

20 Let us first consider :

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23 Optimality


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