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We present a new algorithm, called MCS-M, for computing minimal triangulations of graphs. Lex-BFS, a seminal algorithm for recognizing chordal graphs, was the genesis for two other classical algorithms: LEX M and MCS. LEX M extends the fundamental concept used in Lex-BFS, resulting in an algorithm that not only recognizes chordality, but also computes a minimal triangulation of an arbitrary graph. MCS simplifies the fundamental concept used in Lex-BFS, resulting in a simpler algorithm for recognizing chordal graphs. The new algorithm MCS-M combines the extension of LEX M with the simplification of MCS, achieving all the results of LEX M in the same time complexity.

Intestinal metaplasia has been proposed to be a precursor lesion of adenocarcinoma in the urinary bladder. CDX2 is a transcription factor that is encoded by a homeotype gene that plays an essential role in the differentiation and proliferation of intestinal epithelial cells. Hepatocyte-specific antigen (Hep) has also been shown to be a useful marker of intestinal metaplasia. Tissues from 46 patients, including 22 cases of intestinal metaplasia of the urinary bladder, 11 cases of typical cystitis glandularis, and 13 cases containing both lesions, were selected and immunohistochemical stains for CDX2, Hep, cytokeratin 20 (CK20), and cytokeratin 7 (CK7) were performed. Nuclear staining for CDX2 was observed in 29 of 35 (83%) cases of intestinal metaplasia of the urinary bladder. In contrast, nuclear staining for CDX2 was not observed in any case of typical cystitis glandularis; however, seven of 24 (29%) cases showed aberrant cytoplasmic expression in a mean of 37% of cells. CK20 was expressed in 28 of 35 (80%) cases of intestinal metaplasia, but was observed in only one of 24 (4%) cases of cystitis glandularis in 15% of cells. CK7 was expressed in only six of 35 (17%) cases of intestinal metaplasia, whereas expression of CK7 was observed in all cases (100%) of typical cystitis glandularis with a mean percentage of positively staining cells of 63%. The mean percentages of positively staining cells in intestinal metaplasia with CDX2, CK20, and CK7 were 55, 49, and 53%, respectively. All examples of both intestinal metaplasia and typical cystitis glandularis were uniformly negative for Hep. In the urinary bladder, intestinal metaplasia and typical cystitis glandularis have sharply contrasting immunoprofiles. Additionally, the absence of Hep staining in intestinal metaplasia of the urinary bladder, despite its morphologic resemblance to normal colonic mucosa and intestinal metaplasia in other organs, may signify the presence of unique metaplastic pathways in the urinary bladder. Keywords: urinary bladder; cystitis glandularis; intestinal metaplasia; CDX2; hepatocyte-specific antigen; cytokeratin 20Cystitis glandularis is a metaplastic alteration of the urothelium in the urinary bladder that is thought to be induced by chronic inflammation or irritation.

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Graph search algorithms have exploited graph extremities, such as the leaves of a tree and the simplicial vertices of a chordal graph. Recently, several well-known graph search algorithms have been collectively expressed as two generic algorithms called MLS and MLSM. In this paper, we investigate the properties of the vertex that is numbered 1 by MLS on a chordal graph and by MLSM on an arbitrary graph. We explain how this vertex is an extremity of the graph. Moreover, we show the remarkable property that the minimal separators included in the neighborhood of this vertex are totally ordered by inclusion

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