Computing the fully optimal spanning tree of an ordered bipolar directed graph

It was previously shown by the authors that a directed graph on a linearly ordered set of edges (ordered graph) with adjacent unique source and sink (bipolar digraph) has a unique fully optimal spanning tree, that satisfies a simple criterion on fundamental cycle/cocycle directions. This result is related to a strengthening of the notion of optimality in linear programming. Furthermore, this result yields, for any ordered graph, a canonical bijection between bipolar orientations and spanning trees with internal activity 1 and external activity 0 in the sense of the Tutte polynomial. This bijection can be extended to all orientations and all spanning trees, yielding the active bijection, presented for graphs in other papers. In this paper, we specifically address the problem of the computation of the fully optimal spanning tree of an ordered bipolar digraph. In contrast with the inverse mapping, built by a straightforward single pass over the edge set, the direct computation is not easy and had previously been left aside. We give two independent constructions. The first one is a deletion/contraction recursion, involving an exponential number of minors. It is structurally significant but it is efficient only for building the whole bijection (i.e., all images) at once. The second one is more complicated and is the main contribution of the paper. It involves just one minor for each edge of the resulting spanning tree, and it is a translation and an adaptation to the case of graphs, in terms of weighted cocycles, of a general geometric linear programming type algorithm, which allows for a polynomial time complexity. (c) 2024 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons .org /licenses /by-nc -nd /4 .0/).