Department of Mathematics

Combinatorics and Graph Theory

  •  Ira Gessel, Brandeis University
  •  Counting graphs with neighborhood restrictions
  •  03/31/2021
  •  3:00 PM - 3:50 PM
  •  Online (virtual meeting) (Virtual Meeting Link)
  •  Bruce E Sagan (bsagan@msu.edu)

A graph is called point-determining (or mating type) if no two vertices have the same neighborhood. An arbitrary graph can be reduced to a point-determining graph by contracting each set of vertices with the same neighborhood to a single vertex, and this decomposition enables us to give a simple exponential generating function for counting point-determining graphs, as accomplished by Ronald Read in 1989. In this talk we will discuss a closely related problem: counting graphs in which no two vertices have complementary neighborhoods. The decomposition approach does not work here. Instead we apply inclusion-exclusion, similarly to its use in rook theory, to obtain a simple exponential generating function for these graphs. We also discuss how this application of inclusion-exclusion is related to Möbius inversion, and how it can be applied to some related problems.

 

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Michigan State University
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