The {\sc Plane Diameter Completion} problem asks, given a plane graph G and
a positive integer d, if it is a spanning subgraph of a plane graph H that
has diameter at most d. We examine two variants of this problem where the
input comes with another parameter k. In the first variant, called BPDC, k
upper bounds the total number of edges to be added and in the second, called
BFPDC, k upper bounds the number of additional edges per face. We prove that
both problems are {\sf NP}-complete, the first even for 3-connected graphs of
face-degree at most 4 and the second even when k=1 on 3-connected graphs of
face-degree at most 5. In this paper we give parameterized algorithms for both
problems that run in O(n3)+22O((kd)2logd)⋅n steps.Comment: Accepted in IPEC 201