Nešetřil and Ossona de Mendez introduced the notion of first order convergence as an attempt to unify the notions of convergence for sparse and dense graphs. It is known that there exist first order convergent sequences of graphs with no limit modeling (an analytic representation of the limit). On the positive side, every first order convergent sequence of trees or graphs with no long path (graphs with bounded tree-depth) has a limit modeling. We strengthen these results by showing that every first order convergent sequence of plane trees (trees with embeddings in the plane) and every first order convergent sequence of graphs with bounded path-width has a limit modeling.
For every d ≥ 3 and k ∈ {2} ∪ [3, ∞), we determine the smallest ε such that every fractional (k + ε)-precoloring of vertices at mutual distance at least d of a graph G with fractional chromatic number equal to k can be extended to a proper fractional (k + ε)-coloring of G. Our work complements the analogous results of Albertson for ordinary colorings and those of Albertson and West for circular colorings.
BACKGROUND: Second cancers are an important cause of mortality and morbidity in long-term survivors of testicular germ cell tumors (TGCTs). Studies on the impact of follow-up for the first tumor on the outcome of second malignancies are lacking. The aim of this study was to study the details of diagnosis of second cancers and the role of focused oncology follow-up. METHODS: Medical records and the electronic database of a tertiary referral center for germ cell neoplasms were searched for second cancers diagnosed in TGCT survivors. In a database of 1057 patients, 63 cases of metachronous second malignancies (26 contralateral testicular cancers and 37 nontesticular cancers) were found in 57 patients. Long-term oncology follow-up consisted of yearly history, physical examination, germ cell tumor markers, and imaging including abdominal computed tomography (CT) scans and chest x-ray. RESULTS: The second malignancies occurred after a medium follow-up of 9.9 years (range, 1.1-33 years) after the diagnosis of the first tumor. Only 17 (27%) of the 63 second tumors were detected by oncology follow-up investigations, and a further 12 (29%) were detected by nononcology physicians during a preplanned clinical visit. In 34 (54%) cases, patients themselves or their relatives initiated a clinical appointment because of symptoms. Follow-up investigations all had low yields for the detection of second malignancies, although CT imaging did detect several cases of cancer at an early stage. CONCLUSIONS: In this retrospective study, most second cancers occurring in long-term TGCT survivors were missed by regular oncology follow-up that included yearly physical examination, tumor marker, and imaging.
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