Planar Manhattan Local Minimal and Critical Networks

The present work is devoted to the investigation of branching extremals, i.e., extremal networks, of the Manhattan length functional. Recall that the Manhattan length of a straight segment in Rn is defined as the sum of the lengths of the segment projections to the Cartesian coordinate axis. The Manhattan length of a curve can be defined as the limit of the Manhattan lengths of polygonal lines inscribed into the curve. The Manhattan length of a network (i.e., of a connected set of curves–edges endowed with a graph structure) is the sum of its edges Manhattan lengths. Traditionally the shortest networks are investigated. In the present work we investigate wider classes of networks: locally shortest (so-called local minimal) networks, and critical networks, i.e., critical points of the Manhattan length functional: see the exact definitions below. Notice that in the case of the Riemannian length functional, local minimal networks are critical points of the length functional, and, if splitting of vertices is permitted, then the reverse statement is also true, see [19]. In other words, the class of local minimal networks and the class of critical networks coincide for the case of the Riemannian length functional. It turns out, that in the case of the Manhattan length functional the class of local minimal networks is wider than the one of critical networks. The main aim of the present work is the description of the difference between these classes. The first works on the shortest networks in the sense of the Manhattan length appeared during the 1960s, see [6], due to the intensive development of electronics and robotics. The interest in the Manhattan length appeared due to the fact that, as a rule, conductors on printed circuits have a form of polygonal lines formed by horizontal and vertical segments, and therefore the Manhattan length of the conductors coincides with their Euclidean length. A similar situation takes place in robotics. Rather, the first systematical investigation of the shortest networks in the sense of the Manhattan length (so-called shortest rectilinear trees) was made by Hanan [9] in 1966, who described some important general properties of such networks. In particular, Hanan showed that among the shortest rectilinear trees there always exists one which is a subset of the union of all vertical and horizontal straight lines passing through the boundary points (the union is called a Hanan lattice). Notice that the edges of the shortest rectilinear tree can be chosen in many different ways without changing the length of the tree. But, starting from the work of Hanan [9] the edges of the shortest trees are traditionally chosen in the form of polygonal lines whose links are parallel to the Cartesian coordinate axis. After 10 years Hwang [13] described the possible structures of the shortest rectilinear trees under the assumption that the given boundary set can be spanned by at least one nondegenerate shortest tree 0. The latter means that the degrees of all the boundary vertices in 0 equal to 1. In particular, the tree 0 has no vertices of degree 2. Hwang proved that in this case the shortest tree has one of the following two possible structures, which are depicted in Figure 1.