Outgoing Edge in the Acyclic Orientation), and the Canonical Ordering for Planar P Long 2 3 F F F

triangular graphs 7] (in which every vertex v; v 6 = v 1 ; v 2 ; v n , has 2 incoming and 1 outgoing edge in the acyclic orientation). Another observation is that the canonical ordering, presented in section 4, gives a simple algorithm to test whether a planar triangular graph is 4-connected. An interesting research eld is to problem of computing a canonical ordering of a 4-connected planar graph such that v i+1 is a neighbor of v i. This would yield a simple algorithm for constructing hamiltonian circuits in 4-connected triangular planar graphs. We leave this question open for the interested reader. 1 2 low(F) high(F) w w 1 w Figure 7: Example of the proof of Theorem 5.1. path of F 1 , starting with edge (w 1 ; w 2), has length 2. Hence w 2 has an outgoing edge to a node of F 1 , and an outgoing edge to w 3. Thus w 2 = low(F 2), with F 2 = left((w 2 ; w 3)). Repeating this argument it follows that w d?1 = low(F d?1), with F d?1 = left((w d?1 ; w d)). However it is easy to see that w d = high(F d?1). This means that one of the two directed paths of F d?1 has length 1. This contradiction proves the claim. When traversing an edge e of P long , we visit either left(e) or right(e) (or both) for the rst time. We assign each edge e to the face F, with e 2 F, which we visit for the rst time now. G 0 has n ? 2 faces. To every face F of G 0 , by the claim, at most one edge e 2 P long is assigned. Hence the longest path from s to t in G has length n ? 1. 2 Visibility(G) can be applied to a general 4-connected planar graph by rst triangulating it. (The triangulation of a 4-connected planar graph is clearly still 4-connected.) Since the worst-case bounds for visibility representation by applying an arbitrary st-numbering is (2n ? 5) (n ? 1) 9, 11], our algorithm reduces the width of the visibility representation by a factor 2 in the case of 4-connected planar graphs. Maybe this approach can be used to obtain better grid bounds in general, by splitting the graph into 4-connected components. Consider …