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# Graph centers, hypergraph degree sequences, and induced-saturation

#### Abstract

The *center* of a graph is the set of vertices whose distance to other vertices is minimal. The *centralizing number * of a graph *G* is the minimum number of additional vertices in any graph *H* where *V(G)* is the center of *H.* Buckley, Miller, and Slater and He and Liu provided infinite families of graphs with each centralizing number. We show the number of graphs with each nonzero centralizing number grows super-exponentially with the number of vertices. We also provide a method of altering graphs without changing the centralizing number and give results about the centralizing number of dense graphs. ^ The *degree sequence* of a (hyper)graph is the list of the number of edges containing each vertex. A *t-switch* replaces * t* edges with *t* new edges while maintaining the same degree sequence. For graphs, it has been repeatedly shown that any realization of a degree sequence can be turned into any other realization by a sequence of 2-switches. However, Gabelman provided an example to show 2-switches are not sufficient for *k*-graphs with k > 2. We classify all pairs of 3-graphs that do not admit a 2-switch but differ by a 3-switch. We use this to provide support that 2-switches and a 3-switch are sufficient for 3-graphs. ^ Given graphs *G* and *H, G* is * H*-saturated if *G* does not contain *H* as a subgraph, but *H* is a subgraph of *G* + *e* for any *e* not in *E(G).* While this is well defined for subgraphs, the similar definition is not well defined for induced subgraphs. To avoid this, Martin and Smith defined the * induced-saturation* number using trigraphs. We show that the induced-saturation number of stars is zero. This implies the existence of graphs that are star induced-saturated. We introduce the parameter indsat*(*n, H*) which is the minimum number of edges in an *H*-induced-saturated graph, when one exists. We provide bounds for indsat*(*n, K*_{ 1,3}) and compute it exactly for infinitely many n.^

#### Subject Area

Mathematics

#### Recommended Citation

Behrens, Sarah Lynne, "Graph centers, hypergraph degree sequences, and induced-saturation" (2015). *ETD collection for University of Nebraska - Lincoln*. AAI3715451.

http://digitalcommons.unl.edu/dissertations/AAI3715451