Packings and coverings of the complete graph with trees

In this thesis, we define the spectrum problem for packings (coverings) of G to be the problem of finding all graphs H such that a maximum G-packing (minimum G- covering) of the complete graph with the leave (excess) graph H exists. The set of achievable leave (excess) graphs in G-packings (G-coverings) of the complete graph is called the spectrum of leave (excess) graphs for G. Then, we consider this problem for trees with up to five edges. We will prove that for any tree T with up to five edges, if the leave graph in a maximum T-packing of the complete graph Kn has i edges, then the spectrum of leave graphs for T is the set of all simple graphs with i edges. In fact, for these T and i and H any simple graph with i edges, we will construct a maximum T-packing of Kn with the leave graph H. We will also show that for any tree T with k ≤ 5 edges, if the excess graph in a minimum T-covering of the complete graph Kn has i edges, then the spectrum of excess graphs for T is the set of all simple graphs and multigraphs with i edges, except for the case that T is a 5-star, for which the graph formed by four multiple edges is not achievable when n = 12.

Exploring the Role of Parental Hearsay when Children Witness a Crime

In cases of potential child abuse, parents may provide hearsay testimony on behalf of a child, retelling events from the child’s perspective. However, according to the limited research that exists, parents may have a negative impact on their child’s memory of an event (Principe, DiPuppo, & Gammel, 2013). In order to gain a better understanding of parental hearsay, parents’ descriptions of information children provided in recorded parent-child discussions were compared to the actual information the children provided in the initial discussion and in a 1-week follow-up interview. Children interviewed by parents were also compared to children interviewed by a trained interviewer. To date, 11 children between the ages of 6-9 years have been assessed. While the current sample size was too small to yield many significant results, graphs and effect sizes suggest there are differences in memory accuracy and completeness between parents and children and across children’s interview condition. Whether hearsay testimony or children’s testimony is preferable may depend on how suggestive the initial parent-child discussion is.