Honors Program

Honors in Mathematics

Date of Award

5-2013

Thesis Professor(s)

Teresa Haynes

Thesis Professor Department

Mathematics and Statistics

Thesis Reader(s)

Robert Gardner, Richard Ignace

Abstract

Placing degree constraints on the vertices of a path allows the definitions of uphill and downhill paths. Specifically, we say that a path π = v1, v2,...vk+1 is a downhill path if for every i, 1 ≤ ik, deg(vi) ≥ deg(vi+1). Conversely, a path π = u1, u2,...uk+1 is an uphill path if for every i, 1 ≤ ik, deg(ui) ≤ deg(ui+1). We investigate graphical parameters related to downhill and uphill paths in graphs. For example, a downhill path set is a set P of vertex disjoint downhill paths such that every vertex vV belongs to at least one path in P, and the downhill path number is the minimum cardinality of a downhill path set of G. For another example, the downhill domination number of a graph G is defined to be the minimum cardinality of a set S of vertices such that every vertex in V lies on a downhill path from some vertex in S. The uphill domination number is defined as expected. We determine relationships among these invariants and other graphical parameters related to downhill and uphill paths. We also give a polynomial time algorithm to find a minimum downhill dominating set and a minimum uphill dominating set for any graph.

Document Type

Honors Thesis - Withheld

Copyright

Copyright by the authors.

Included in

Mathematics Commons

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