Consider a convex polygon P in the plane, and denote by U a homothetical copy of the vector sum of P and −P . Then the polygon U , as unit ball, induces a norm such that, with respect to this norm, P has constant Minkowskian width. We define notions like Minkowskian curvature, evolutes and involutes for polygons of constant U -width, and we prove that many properties of the smooth case, which i...

background: considering the effectiveness of craniofacial photographic analysis for diagnosis and management of obstructive sleep apnea-hypopnea syndrome (osahs) as well as ethnic differences in indexes measured by this method, this study designed to compare the surface facial dimensions, including nose width, intercanthal width and mandibular width of iranian patients with mild, moderate and s...

The k-power graph of a graph G is a graphwith the same vertex set as G, in that two vertices are adjacent if and only if, there is a path between them in G of length at most k. A k-treepower graph is the k-power graph of a tree, a k-leaf-power graph is the subgraph of some k-tree-power graph induced by the leaves of the tree. We show that (1) every k-tree-power graph has NLC-width at most k + 2...

Let tw(G), pw(G), c(G), !J.(G) denote, respectively, the tree-width, path-width, cutwidth and the maximum degree of a graph G on 11 vertices . It is known that c (G)~tw (G). We prove that c (G) =0 (tw (G)·!J.(G)·logn), and if ({Xj : iel] ,T=(I,A» is a tree decomposition of G with tree-wid~ then c (G) S (k+ l)·!J.(G)·c (T). In case that a tree decomposition is given, or that the tree-width is bo...