#### Degree Name

MS (Master of Science)

#### Program

Mathematical Sciences

#### Date of Award

5-2009

#### Committee Chair or Co-Chairs

Robert A. Beeler

#### Committee Members

Robert B. Gardner, Teresa W. haynes

#### Abstract

A decomposition *D* of a graph *H* by a graph *G* is a partition of the edge set of *H* such that the subgraph induced by the edges in each part of the partition is isomorphic to *G*. A *mixed graph* on *V* vertices is an ordered pair (*V*,*C*), where *V* is a set of vertices, |*V*| = *v*, and *C* is a set of ordered and unordered pairs, denoted (*x*, *y*) and [*x*, *y*] respectively, of elements of *V* [8]. An ordered pair (*x*, *y*) ∈ *C* is called an *arc* of (*V*,*C*) and an unordered pair [*x*, *y*] ∈ *C* is called an *edge* of graph (*V*,*C*). A path on *n* vertices is denoted as *P _{n}*. A

*partial orientation*on

*G*is obtained by replacing each edge [

*x*,

*y*] ∈

*E*(

*G*) with either (

*x*,

*y*), (

*y*,

*x*), or [

*x*,

*y*] in such a way that there are twice as many arcs as edges. The

*complete mixed graph*on

*v*vertices, denoted

*M*, is the mixed graph (

_{v}*V*,

*C*) where for every pair of distinct vertices

*v*

_{1},

*v*

_{2}∈

*V*, we have {(

*v*

_{1},

*v*

_{2}), (

*v*

_{2},

*v*

_{1}), [

*v*

_{1},

*v*

_{2}]} ⊂

*C*. The goal of this thesis is to establish necessary and sufficient conditions for decomposition of

*M*by all possible partial orientations of

_{v}*P*

_{4}.

#### Document Type

Thesis - Open Access

#### Recommended Citation

Meadows, Adam M., "Decompositions of Mixed Graphs with Partial Orientations of the P_{4}." (2009). *Electronic Theses and Dissertations.* Paper 1870. http://dc.etsu.edu/etd/1870

#### Copyright

Copyright by the authors.