No abstract

Every graph generated by a hyperedge replacement graph-grammar can be represented by a tree, namely the derivation tree of the derivation sequence that produced it. Certain functions on graphs can be computed recursively on the derivation trees of these graphs. By using monadic second-order logic and semiring homomorphisms, we describe in a single formalism a large class of such functions. Polynomial and even linear algorithms can be constructed for some of these functions. We unify similar results obtained by Takamizawa et al. (1982), Bern et al. (1987), Arnborg et al. (1991) and Habel et al. (1989).

We use Schnyder woods of 3-connected planar graphs to produce convex straight line drawings on a grid of size (n − 2 − ∆) × (n − 2 − ∆). The parameter ∆ ≥ 0 depends on the Schnyder wood used for the drawing. This parameter is in the range 0The algorithm is a refinement of the face-counting-algorithm, thus, in particular, the size of the grid is at most (f − 2) × (f − 2).The above bound on the grid size simultaneously matches or improves all previously known bounds for convex drawings, in particular Schnyder's and the recent Zhang and He bound for triangulations and the Chrobak and Kant bound for 3-connected planar graphs. The algorithm takes linear time.The drawing algorithm has been implemented and tested. The expected grid size for the drawing of a random triangulation is close to