We introduce a new chamfering paradigm, locally connecting pixels to produce path distances that approximate Euclidean space by building a small network (a replacement product) inside each pixel. These " RE -grid graphs" maintain near-Euclidean polygonal distance contours even in noisy data sets, making them useful tools for approximation when exact numerical solutions are unobtainable or impractical. The RE -grid graph creates a modular global architecture with lower pixel-to-pixel valency and simplified topology at the cost of increased computational complexity due to its internal structure. We present an introduction to chamfering replacement products with a number of case study examples to demonstrate the potential of these graphs for path-finding in high frequency and low resolution image spaces which motivate further study. Possible future applications include morphology, watershed segmentation, halftoning, neural network design, anisotropic image processing, image skeletonization, dendritic shaping, and cellular automata.

In this paper, we construct new families of graphs whose automorphism groups are transitive on 3-paths. These graphs are constructed from certain Lie algebras related to generalized Kac--Moody algebras of rank two. We will show that one particular subfamily gives new lower bounds on the number of edges in extremal graphs with no cycles of length fourteen.