Resolving Prime Modules: The Structure of Pseudo-cographs and Galled-Tree Explainable Graphs

The modular decomposition of a graph $G$ is a natural construction to capture key features of $G$ in terms of a labeled tree $(T,t)$ whose vertices are labeled as "series" ($1$), "parallel" ($0$) or "prime". However, full information of $G$ is provided by its modular decomposition tree $(T,t)$ only, if $G$ is a cograph, i.e., $G$ does not contain prime modules. In this case, $(T,t)$ explains $G$, i.e., $\{x,y\}\in E(G)$ if and only if the lowest common ancestor $\mathrm{lca}_T(x,y)$ of $x$ and $y$ has label "$1$". Pseudo-cographs, or, more general, GaTEx graphs $G$ are graphs that can be explained by labeled galled-trees, i.e., labeled networks $(N,t)$ that are obtained from the modular decomposition tree $(T,t)$ of $G$ by replacing the prime vertices in $T$ by simple labeled cycles. GaTEx graphs can be recognized and labeled galled-trees that explain these graphs can be constructed in linear time. In this contribution, we provide a novel characterization of GaTEx graphs in terms of a set $\mathfrak{F}_{\mathrm{GT}}$ of 25 forbidden induced subgraphs. This characterization, in turn, allows us to show that GaTEx graphs are closely related to many other well-known graph classes such as $P_4$-sparse and $P_4$-reducible graphs, weakly-chordal graphs, perfect graphs with perfect order, comparability and permutation graphs, murky graphs as well as interval graphs, Meyniel graphs or very strongly-perfect and brittle graphs. Moreover, we show that every GaTEx graph as twin-width at most 1.Comment: 18 pages, 3 figure

Summary and Outlook for 9th International Symposium on Heavy Flavor Physics

This is the summary talk of a meeting held at the California Institute of Technology Sept 10-13, 2001. I do not attempt to summarize all the beautiful experimental results we have seen this week, nor to repeat the lively theoretical discussions that have occurred. Rather I will present my own biased perspective on what we have learned, and on the important tasks that need our attention as we work to make the most of the rapidly accumulating data in this field.Comment: Talk presented at 9th International Symposium on Heavy Flavor Physics, California Institute of Technology, September 10-13, 200

Decomposition of Pig Carcasses at Varying Room Temperature

The goal of this study is to assess the qualitative scoring of decomposition scales with statistical analysis while also determining which indoor environment will decompose the fastest. When comparing the three indoor environments, water decomposition appears to be the fastest, but the method of analysis for this study determines that using decomposition scales to analyze such environments may produce statistically insignificant results. For this study, piglets were put in a dry steel tub, a water filled steel tub, and a suitcase. These were recorded using photographs for 25 days and then scored using decomposition scales. The total score was divided by all points possible to show a percentage of decomposition and was compared between three different scales using statistical analysis. Therefore, relying on decomposition scales as the only forensic analysis for decomposition to observe how different environments affect cadavers may be problematic, as bodies found indoors tend to be in the early stages of decomposition. The statistical insignificance of the total body score further suggests decomposition scales and postmortem interval methods that use these scales should not be used alone for courtroom purposes, and medicolegal investigators need to apply other scientific methods in order to support their postmortem interval theory