An eternal $m$-secure set of a graph $G = (V,E)$ is aset $S_0subseteq V$ that can defend against any sequence ofsingle-vertex attacks by means of multiple-guard shifts along theedges of $G$. A suitable placement of the guards is called aneternal $m$-secure set. The eternal $m$-security number$sigma_m(G)$ is the minimum cardinality among all eternal$m$-secure sets in $G$. An edge $uvin E(G)$ is ...

Let Xn := {xi}ni=0 be a given set of (n + 1) pairwise distinct points in R (called nodes or sample points), let P = conv(Xn), let f be a convex function with Lipschitz continuous gradient on P and λ := {λi}ni=0 be a set of barycentric coordinates with respect to the point set Xn. We analyze the error estimate between f and its barycentric approximation:

A bipartite graph G = (X,Y ;E) is convex if there exists a linear enumeration L of the vertices of X such that the neighbours of each vertex of Y are consecutive in L. We show that the problems of finding a minimum dominating set and a minimum independent dominating set in an n-vertex convex bipartite graph are solvable in time O(n2). This improves previous O(n3) algorithms for these problems. ...