x1 is the geodesic path from x2 to x3 via or through points including/on connected manifold component/in connected graph component x4, with distance being measured by standard/metric/weighting x5.
x1 and x2 must belong to the same connected component (which includes/encompasses x4). The overall 'distance' travelled must be the minimal option under the orientation (from x2 to x3; this is not always symmetric) and path weighting/the specified metric such that the points x4 are included in the path or the path is in/on connected manifold/graph component x4 (as appropriate). In other words, x1 includes a geodesic from x2 to x3 through/on/along/in points or space x4. If the notion of distance is well-formed, then the triangle inequality should be satisfied and, therefore, x1 should be composed of certain of geodesic subpaths between points in x4 U Set(x2, x3). If x4 is oriented or is an ordered list of points, then the path must connect them in that order (or a permitted order if there are multiple options).