Separability and Vertex Ordering of Graphs

Many graph optimization problems, such as finding an optimal coloring, or a largest clique, can be solved by a divide-and-conquer approach. One such well-known technique is decomposition by clique separators where a graph is decomposed into special induced subgraphs along their clique separators. While the most common practice of this method employs minimal clique separators, in this work we study other variations as well. We strive to characterize their structure and in particular the bound on the number of atoms. In fact, we strengthen the known bounds for the general clique cutset decomposition and the minimal clique separator decomposition. Graph ordering is the arrangement of a graph’s vertices according to a certain logic and is a useful tool in optimization problems. Special types of vertices are often recognized in graph classes, for instance it is well-known every chordal graph contains a simplicial vertex. Vertex-ordering, based on such properties, have originated many linear time algorithms. We propose to define a new family named SE-Class such that every graph belonging to this family inherently contains a simplicial extreme, that is a vertex which is either simplicial or has exactly two neighbors which are non-adjacent. Our family lends itself to an ordering based on simplicial extreme vertices (named SEO) which we demonstrate to be advantageous for the coloring and maximum clique problems. In addition, we examine the relation of SE-Class to the family of (Even-Hole, Kite)-free graphs and show a linear time generation of SEO for (Even-Hole, Diamond, Claw)-free graphs. We showcase the applications of those two core tools, namely clique-based decomposition and vertex ordering, on the (Even-Hole, Kite)-free family

Separability and Vertex Ordering of Graphs

Many graph optimization problems, such as finding an optimal coloring, or a largest clique, can be solved by a divide-and-conquer approach. One such well-known technique is decomposition by clique separators where a graph is decomposed into special induced subgraphs along their clique separators. While the most common practice of this method employs minimal clique separators, in this work we study other variations as well. We strive to characterize their structure and in particular the bound on the number of atoms. In fact, we strengthen the known bounds for the general clique cutset decomposition and the minimal clique separator decomposition. Graph ordering is the arrangement of a graph’s vertices according to a certain logic and is a useful tool in optimization problems. Special types of vertices are often recognized in graph classes, for instance it is well-known every chordal graph contains a simplicial vertex. Vertex-ordering, based on such properties, have originated many linear time algorithms. We propose to define a new family named SE-Class such that every graph belonging to this family inherently contains a simplicial extreme, that is a vertex which is either simplicial or has exactly two neighbors which are non-adjacent. Our family lends itself to an ordering based on simplicial extreme vertices (named SEO) which we demonstrate to be advantageous for the coloring and maximum clique problems. In addition, we examine the relation of SE-Class to the family of (Even-Hole, Kite)-free graphs and show a linear time generation of SEO for (Even-Hole, Diamond, Claw)-free graphs. We showcase the applications of those two core tools, namely clique-based decomposition and vertex ordering, on the (Even-Hole, Kite)-free family

Risk of prostate cancer for men with prior negative biopsies undergoing magnetic resonance imaging compared with biopsy-naive men: A prospective evaluation of the PLUM cohort.

BACKGROUND: Men with prior negative prostate biopsies have a lower risk of being diagnosed with prostate cancer in comparison with biopsy-naive men. However, the relative clinical utility of identified lesions on multiparametric magnetic resonance imaging (mpMRI) is uncertain between the 2 settings. METHODS: Patients from the Prospective Loyola University mpMRI (PLUM) Prostate Biopsy Cohort (January 2015 to June 2020) were examined. The detection of any prostate cancer and clinically significant prostate cancer (Gleason score ≥ 3 + 4) was stratified by Prostate Imaging-Reporting and Data System (PI-RADS) scores in the prior negative and biopsy-naive settings. Multivariable logistic regression models (PLUM models) assessed predictors, and decision curve analyses were used to estimate the clinical utility of PI-RADS cutoffs relative to the models. RESULTS: Nine hundred men (420 prior negative patients and 480 biopsy-naive patients) were included. Prior negative patients had lower risks of any prostate cancer (27.9% vs 54.4%) and clinically significant prostate cancer (20.0% vs 38.3%) in comparison with biopsy-naive patients, and this persisted when they were stratified by PI-RADS (eg, PI-RADS 3: 13.6% vs 27.4% [any prostate cancer] and 5.2% vs 15.4% [clinically significant prostate cancer]). The rate of detection of clinically significant prostate cancer was 5.3% among men with prior negative biopsy and PI-RADS ≤ 3. Family history and Asian ancestry were significant predictors among biopsy-naive patients. PLUM models demonstrated a greater net benefit and reduction in biopsies (45.8%) without missing clinically significant cancer in comparison with PI-RADS cutoffs (PI-RADS 4: 34.0%). CONCLUSIONS: Patients with prior negative biopsies had lower prostate cancer detection by PI-RADS score category in comparison with biopsy-naive men. Decision curve analyses suggested that many biopsies could be avoided by the use of the PLUM models or a PI-RADS 4 cutoff without any clinically significant cancer being missed. LAY SUMMARY: Men with a prior negative prostate biopsy had a lower risk of harboring prostate cancer in comparison with those who never had a biopsy. This was true even when patients in each group had similar multiparametric magnetic resonance imaging (mpMRI) findings in terms of Prostate Imaging-Reporting and Data System (PI-RADS)-graded lesions. Decision curve analyses showed that many biopsies could be avoided by the use of the Prospective Loyola University mpMRI prediction models or a PI-RADS 4 cutoff for patients with prior negative biopsies

Systematic versus Targeted Magnetic Resonance Imaging/Ultrasound Fusion Prostate Biopsy among Men with Visible Lesions.

PURPOSE: Multiparametric magnetic resonance imaging (mpMRI)-ultrasound (US) fusion-guided biopsy may improve prostate cancer (PCa) detection and reduce grade misclassification. We compared PCa detection rates on systematic, magnetic resonance imaging-targeted, and combined biopsy with evaluation of important subgroups. MATERIALS AND METHODS: Men with clinical suspicion of harboring PCa from 2 institutions with visible Prostate Imaging-Reporting and Data System (PI-RADS RESULTS: Of 1,236 patients (647 biopsy-naïve) included, 626 (50.6%) harbored PCa and 412 (33.3%) had csPCa on combined biopsy. Detection of csPCa was 27.9% vs 23.3% (+4.6%) and GG1 PCa was 11.3% vs 17.8% (-6.5%) for targeted vs systematic cores. Benefit in csPCa detection was higher in the prior negative than biopsy-naïve setting (+7.8% [p CONCLUSIONS: Combined biopsy doubled the benefit of targeted biopsy alone in detection of csPCa without increasing GG1 PCa diagnoses relative to systematic biopsy. Utility of targeted biopsy was higher in the prior negative biopsy cohort, but advantages of combined biopsy were maintained regardless of biopsy history

A prostate biopsy risk calculator based on MRI: development and comparison of the Prospective Loyola University multiparametric MRI (PLUM) and Prostate Biopsy Collaborative Group (PBCG) risk calculators.

OBJECTIVES: To develop and validate a prostate cancer (PCa) risk calculator (RC) incorporating multiparametric magnetic resonance imaging (mpMRI) and to compare its performance with that of the Prostate Biopsy Collaborative Group (PBCG) RC. PATIENTS AND METHODS: Men without a PCa diagnosis receiving mpMRI before biopsy in the Prospective Loyola University mpMRI (PLUM) Prostate Biopsy Cohort (2015-2020) were included. Data from a separate institution were used for external validation. The primary outcome was diagnosis of no cancer, grade group (GG)1 PCa, and clinically significant (cs)PCa (≥GG2). Binary logistic regression was used to explore standard clinical and mpMRI variables (prostate volume, Prostate Imaging-Reporting Data System [PI-RADS] version 2.0 lesions) with the final PLUM RC, based on a multinomial logistic regression model. Receiver-operating characteristic curve, calibration curves, and decision-curve analysis were evaluated in the training and validation cohorts. RESULTS: A total of 1010 patients were included for development (N = 674 training [47.8% PCa, 30.9% csPCa], N = 336 internal validation) and 371 for external validation. The PLUM RC outperformed the PBCG RC in the training (area under the curve [AUC] 85.9% vs 66.0%; P \u3c 0.001), internal validation (AUC 88.2% vs 67.8%; P \u3c 0.001) and external validation (AUC 83.9% vs 69.4%; P \u3c 0.001) cohorts for csPCa detection. The PBCG RC was prone to overprediction while the PLUM RC was well calibrated. At a threshold probability of 15%, the PLUM RC vs the PBCG RC could avoid 13.8 vs 2.7 biopsies per 100 patients without missing any csPCa. At a cost level of missing 7.5% of csPCa, the PLUM RC could have avoided 41.0% (566/1381) of biopsies compared to 19.1% (264/1381) for the PBCG RC. The PLUM RC compared favourably with the Stanford Prostate Cancer Calculator (SPCC; AUC 84.1% vs 81.1%; P = 0.002) and the MRI-European Randomized Study of Screening for Prostate Cancer (ERSPC) RC (AUC 84.5% vs 82.6%; P = 0.05). CONCLUSIONS: The mpMRI-based PLUM RC significantly outperformed the PBCG RC and compared favourably with other mpMRI-based RCs. A large proportion of biopsies could be avoided using the PLUM RC in shared decision making while maintaining optimal detection of csPCa