# Caterpillar tolerance representations of graphs

#### Abstract

A caterpillar, *H*, is a tree containing a path, * P*, such that every vertex of *H* is either in * P* or adjacent to *P*. Given a graph *G*, a caterpillar tolerance representation of tolerance *t* on a caterpillar *H*, is a mapping of each vertex *v* ∈ * V*(*G*) to a subtree *H _{v}* ⊆

*H*, such that

*uv*∈

*E*(

*G*) if and only if

*H*and

_{u}*H*share at least

_{ v}*t*vertices. ^ Denote by cat[h,t] the set of all graphs for which there exists a tolerance

*t*representation of

*G*in a caterpillar of maximum degree

*h*. Given any positive integers,

*h*and

*t*, it is determined for all values of

*n*whether or not a caterpillar tolerance representations exists for a cycle of length

*n*using a representing caterpillar of maximum degree

*h*and tolerance

*t*. ^ We show equivalence among various classes including cat[3,1] = cat[h,1] = cat[3,2] for

*h*≥ 3 and cat[4,2] = cat[3,3]. Also, for

*h*≥ 5 we show that cat[

*h*,2] ⊊︀ cat[

*h*− 1,3]. ^ A vertex,

*v*, in a graph,

*G*, is called

*simplicial*if the set of neighbors of

*v*induces a clique in

*G*. We say that a graph

*G*has an

*asimplicial asteroidal triple*if there exists in

*G*three distinct vertices

*v*

_{1},

*v*

_{2},

*v*

_{3}, none of which are simplicial, and such that for all permutations

*i, j, k*of the indices 1,2,3 there exits a

*v*path which is distance at least two from

_{i}v_{j}*v*. We provide the characterization of cat[3,1] as the set of all chordal graphs that do not contain an

_{k}*asimplicial asteroidal triple*. And since cat[3,1] = cat[h,1] = cat[3,2] for

*h*≥ 3, we get characterizations for them as well. ^

#### Subject Area

Mathematics

#### Recommended Citation

Glenn E Faubert,
"Caterpillar tolerance representations of graphs"
(2005).
*Dissertations and Master's Theses (Campus Access).*
Paper AAI3186904.

http://digitalcommons.uri.edu/dissertations/AAI3186904