A Strongly-Uniform Slicewise Polynomial-Time Algorithm for the Embedded Planar Diameter Improvement Problem
International Symposium on Parameterized and Exact Computation
In the embedded planar diameter improvement problem (EPDI) we are given a graph G embedded in the plane and a positive integer d. The goal is to determine whether one can add edges to the planar embedding of G in such a way that planarity is preserved and in such a way that the resulting graph has diameter at most d. Using non-constructive techniques derived from Robertson and Seymour's graph minor theory, together with the effectivization by self-reduction technique introduced by Fellows and
... ngston, one can show that EPDI can be solved in time 1 ) for some function f (d). The caveat is that this algorithm is not strongly uniform in the sense that the function f (d) is not known to be computable. On the other hand, even the problem of determining whether EPDI can be solved in time f 1 (d) • |V (G)| f2 (d) for computable functions f 1 and f 2 has been open for more than two decades [Cohen at. al. Journal of Computer and System Sciences, 2017]. In this work we settle this later problem by showing that EPDI can be solved in time f (d) • |V (G)| O(d) for some computable function f . Our techniques can also be used to show that the embedded k-outerplanar diameter improvement problem (k-EOPDI), a variant of EPDI where the resulting graph is required to be k-outerplanar instead of planar, can be solved in time f (d) • |V (G)| O(k) for some computable function f . This shows that for each fixed k, the problem k-EOPDI is strongly uniformly fixed parameter tractable with respect to the diameter parameter d.