Ogden, Shannon

Project title: Total Roman Domination Edge-Critical Trees

Department: Mathematics and Statistics

Faculty supervisor: Dr. Kieka Mynhardt

"A Roman dominating function on a graph G is a function f assigning a value of 0,1 or 2 to each vertex in G, such that every vertex which is assigned a 0 is adjacent to some vertex that is assigned a 2. If the subgraph induced by the positive vertices is isolate-free, then f is a total Roman dominating function (abbreviated TRD-function). The total Roman domination number (abbreviated TRD-number) is the minimum weight of a TRD-function on G. The addition of an edge to a graph has the potential to change its TRD-number. A graph G is total Roman domination edge-critical if the addition of any edge to G decreases its TRD-number. In this project, we will investigate the properties of various total Roman domination edge-critical trees, beginning with spiders and double-spiders, with the goal of characterizing which trees are total Roman domination edge-critical."