# Some results on tree tolerance representations

#### Abstract

Consider a simple undirected graph *G* = (* V, E*). A family of subtrees, {*S _{v}*}

_{ v∈V}, of a host tree

*H*is called a

*t*-subtree representation of

*G*provided

*uv*∈

*E*if and only if :

*V*(

*S*) ∩

_{u}*V*(

*S*): ≥

_{v}*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 (

*H*)-representations for cycles and trees, where

_{m}, t*H*is the host tree with exactly one node

_{m}*v*, of degree three and exactly three leaves, each with distance

*m*from

*v*. We denote the set of (

*H*)-representable graphs for some positive integer

_{m}, t*m*, as H (

*t*). We derive an upper and lower bound for the maximum size of a cycle in H (

*t*). Then we characterize the set of all trees in H (

*t*) for

*t*= 1, 2 and 3. In addition, we determine a necessary condition for a graph

*G*to be in H (1) and conjecture that this condition is also sufficient. In our last result we show that any graph

*G*in H (

*t*) is also in H (

*t*+ 1) for all

*t*. ^

#### Subject Area

Mathematics

#### Recommended Citation

Mary Ann Saadi,
"Some results on tree tolerance representations"
(2001).
*Dissertations and Master's Theses (Campus Access).*
Paper AAI3039083.

http://digitalcommons.uri.edu/dissertations/AAI3039083