Triangulating Planar Graphs While Minimizing the Maximum Degree

Triangulating Planar Graphs While Minimizing the Maximum Degree

Triangulating Planar Graphs While Keeping the Pathwidth Small

Any simple planar graph can be triangulated, i.e., we can add edges to it, without adding multi-edges, such that the result is planar and all faces are triangles. In this paper, we study the problem of triangulating a planar graph without increasing the pathwidth by much. We show that if a planar graph has pathwidth k, then we can triangulate it so that the resulting graph has pathwidth O(k) (w...

The maximum degree of random planar graphs

McDiarmid and Reed [‘On the maximum degree of a random planar graph’, Combin. Probab. Comput. 17 (2008) 591–601] showed that the maximum degree Δn of a random labeled planar graph with n vertices satisfies with high probability (w.h.p.) c1 logn < Δn < c2 logn, for suitable constants 0 < c1 < c2. In this paper, we determine the precise asymptotics, showing in particular that w.h.p. |Δn − c logn|...

The maximum degree of planar graphs I. Series-parallel graphs

We prove that the maximum degree ∆n of a random series-parallel graph with n vertices satisfies ∆n/ log n → c in probability, and E∆n ∼ c log n for a computable constant c > 0. The same result holds for outerplanar graphs.

On the maximum degree of path-pairable planar graphs

A graph is path-pairable if for any pairing of its vertices there exist edge-disjoint paths joining the vertices in each pair. We investigate the behaviour of the maximum degree in path-pairable planar graphs. We show that any n-vertex path-pairable planar graph must contain a vertex of degree linear in n.