Some results on tree tolerance representations

Author

Mary Ann Saadi, University of Rhode Island

Date of Award

2001

Degree Type

Dissertation

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.

Recommended Citation

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