Minimal split completions

We study the problem of adding an inclusion minimal set of edges to a given arbitrary graph so that the resulting graph is a split graph, called a minimal split completion of the input graph. Minimal completions of arbitrary graphs into chordal and interval graphs have been studied previously, and new results have been added recently. We extend these previous results to split graphs by giving a linear-time algorithm for computing minimal split completions. We also give two characterizations of minimal split completions, which lead to a linear time algorithm for extracting a minimal split completion from any given split completion.We prove new properties of split graph that are both useful for our algorithms and interesting on their own. First, we present a new way of partitioning the vertices of a split graph uniquely into three subsets. Second, we prove that split graphs have the following property: given two split graphs on the same vertex set where one is a subgraph of the other, there is a sequence of edges that can be removed from the larger to obtain the smaller such that after each edge removal the modified graph is split.     

Uncovering a CF3 Effect on X-ray Absorption Energies of [Cu(CF3)4]− and Related Copper Compounds by Using Resonant Diffraction Anomalous Fine Structure (DAFS) Measurements**

Understanding the electronic structures of high-valent metal complexes aids the advancement of metal-catalyzed cross coupling methodologies. A prototypical complex with formally high valency is [Cu(CF₃)₄]⁻ ( 1 ), which has a formal Cu(III) oxidation state but whose physical analysis has led some to a Cu(I) assignment in an inverted ligand field model. Recent examinations of 1 by X-ray spectroscopies have led previous authors to contradictory conclusions, motivating the re-examination of its X-ray absorption profile here by a complementary method, resonant diffraction anomalous fine structure (DAFS). From analysis of DAFS measurements for a series of seven mononuclear Cu complexes including 1 , here it is shown that there is a systematic trifluoromethyl effect on X-ray absorption that blue shifts the resonant Cu K-edge energy by 2–3 eV per CF₃, completely accounting for observed changes in DAFS profiles between formally Cu(III) complexes like 1 and formally Cu(I) complexes like (Ph₃P)₃CuCF₃ ( 3 ). Thus, in agreement with the inverted ligand field model, the data presented herein imply that 1 is best described as containing a Cu(I) ion with dⁿ count approaching 10.