2-Connecting outerplanar graphs without blowing up the pathwidth

Abstract

Given a connected outerplanar graph G of pathwidth p, we give an algorithm to add edges to G to get a supergraph of G, which is 2-vertex-connected, outerplanar and of pathwidth O(p). This settles an open problem raised by Biedl [1], in the context of computing minimum height planar straight line drawings of outerplanar graphs, with their vertices placed on a two dimensional grid. In conjunction with the result of this paper, the constant factor approximation algorithm for this problem obtained by Biedl [1] for 2-vertex-connected outerplanar graphs will work for all outer planar graphs.

Height-Preserving Transformations of Planar Graph Drawings

“…2(left). Hence it requires width at least 1 3 2 d by the previous lemma and Γ H cannot be smaller. One can easily show that H requires at least 4 rows in any straight-line drawing.…”

“…The height is thus measured by the number of rows, i.e., horizontal lines with integer y-coordinates that intersect the drawing. 1 Drawings in this paper are required to be grid-drawings, with some exceptions for ease of description that will be pointed out as they occur.…”

A polynomial-time algorithm for Outerplanar Diameter Improvement

“…In contrast, the network topology, such as acyclicity or planarity, as a constraint to be preserved has received little attention in the context of complementing a graph; see for example [14]. See also [20,21] for other augmentation problems in outerplanar graphs, where the objective is to add edges in order to achieve a prescribed connectivity.…”

Triangulating Planar Graphs While Keeping the Pathwidth Small

“…(For if so, then all outer-planar graphs can be drawn with height O(pw(G)).) This question was answered in the affirmative by Babu et al [1], who showed that any outer-planar graph G can be made into a 2-connected outer-planar graph G with pw(G ) ≤ 16pw(G) + 15.…”

“…Research was supported by NSERC and done while visiting Universität Salzburg. Many thanks to Jasine Babu for sharing her manuscript of what later became [1], and the referees of an earlier version of this paper for helpful comments.…”