Some results on tree tolerance representations

Author

Mary Ann Saadi, University of Rhode Island

Date of Award

2001

Degree Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

First Advisor

Nancy Eaton

Abstract

Consider a simple undirected graph G = ( V, E). A family of subtrees, {Sv} v∈V, of a host tree H is called a t-subtree representation of G provided uv ∈ E if and only if |V(Su) ∩ V(Sv)| ≥ t. An ( H, t)-representation of G is a t-subtree representation of G with fixed host tree H. In this dissertation we will consider (Hm, t)-representations for cycles and trees, where Hm is the host tree with exactly one node v, of degree three and exactly three leaves, each with distance m from v. We denote the set of (Hm, t)-representable graphs for some positive integer m, as [special characters omitted](t). We derive an upper and lower bound for the maximum size of a cycle in [special characters omitted](t). Then we characterize the set of all trees in [special characters omitted](t) for t = 1, 2 and 3. In addition, we determine a necessary condition for a graph G to be in [special characters omitted](1) and conjecture that this condition is also sufficient. In our last result we show that any graph G in [special characters omitted](t) is also in [special characters omitted](t + 1) for all t.

Comments

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Recommended Citation

Saadi, Mary Ann, "Some results on tree tolerance representations" (2001). Open Access Dissertations. Paper 1793.
https://digitalcommons.uri.edu/oa_diss/1793