Recognizing Visibility Graphs of Polygons with Holes and Internal-External Visibility Graphs of Polygons

Visibility graph of a polygon corresponds to its internal diagonals and boundary edges. For each vertex on the boundary of the polygon, we have a vertex in this graph and if two vertices of the polygon see each other there is an edge between their corresponding vertices in the graph. Two vertices of a polygon see each other if and only if their connecting line segment completely lies inside the polygon, and they are externally visible if and only if this line segment completely lies outside the polygon. Recognizing visibility graphs is the problem of deciding whether there is a simple polygon whose visibility graph is isomorphic to a given input graph. This problem is well-known and well-studied, but yet widely open in geometric graphs and computational geometry. Existential Theory of the Reals is the complexity class of problems that can be reduced to the problem of deciding whether there exists a solution to a quantifier-free formula F(X1,X2,...,Xn), involving equalities and inequalities of real polynomials with real variables. The complete problems for this complexity class are called Existential Theory of the Reals Complete. In this paper we show that recognizing visibility graphs of polygons with holes is Existential Theory of the Reals Complete. Moreover, we show that recognizing visibility graphs of simple polygons when we have the internal and external visibility graphs, is also Existential Theory of the Reals Complete.Comment: Sumbitted to COCOON2018 Conferenc

Diffusion-weighted magnetic resonance imaging at 1.5 T for peripheral zone prostate cancer : the influence of the b-value combination on the diagnostic performance of apparent diffusion coefficient

Purpose: Diffusion-weighted imaging as a noninvasive functional modality plays a valuable role in the evaluation of prostate cancer. However, there is still no agreement on the number and range of b-values to be used. Therefore, the purpose of this study is to investigate the influence of b-value choice on the diagnostic performance of apparent diffusion coefficient (ADC) values for prostate cancer detection. Material and methods: Fifty-nine consecutive patients with abnormal digital rectal examination findings and raised serum prostate-specific antigen were chosen for magnetic resonance imaging of the prostate before systematic 12-core transrectal ultrasound-guided prostate biopsies. ADC values for each ROI were calculated from different b-value combinations (0-1600 s/mm2) by a monoexponential model. Mann-Whitney and the paired-sample t-test were used to compare the mean ADC values for malignant lesions and noncancerous tissues. ROC curve analysis was used to evaluate the diagnostic performance of ADC values in distinguishing prostate cancer from normal-tissue ROIs. Results: The differences between mean ADC values of malignant lesions and contralateral healthy tissues were significant for all the pairs of b-value combinations. The pair of b-values 50 and 1200 provided the highest AUC (0.94), with a sensitivity of 90.2%, a specificity of 92.6%, and an accuracy of 91.2% at an ADC cut-off of 1.23 × 10-3 mm2/s. Conclusions: Our study showed that using a 1.5-Tesla MRI scanner the diagnostic performance of ADC values estimated from the b-value pair 50 and 1200 s/mm2 was highest. However, some other b-value pairs provided statically comparable diagnostic performance