Braced edges in plane triangulations

A plane triangulation is an embedding of a maximal planar graph in the Euclidean plane. Foulds and Robinson (1979) first studied the problem of transforming one triangulation to another by a sequence of diagonal operations. where a diagonal operation deletes one edge and inserts the other diagonal of the resulting quadrilateral face. An edge which cannot be removed by a single diagonal operation is called braced. This paper is a study of the possible number and distribution of braced edges in a triangulation. It shows that at most 2n-4 edges of a triangulation of order n can be braced, and that for any r ::; 2n-4 (with exactly one exception) there is a plane triangulation of order n with r braced edges, so long as n is large enough. 1. What are Braced Edges? A plane triangulation T is an embedding of a maximal planar graph in the euclidean plane. The triangulation T is of order n if it has n vertices, and then Euler's polyhedral fonnula shows it has 3n-6 edges and 2n-4 faces, all triangles (that is, regions bounded by three vertices and three edges). Suppose T has order n2:4. Then with each edge vw of T we can associate a pair of distinct vertices {x,y}, where each is the third vertex of a face incident with vw. If T contains an edge xy, we say that vw is braced, and xy is the edge which braces vw; if there is no edge incident with both x and y, we say that vw is unbraced (Figure 1), If FIGURE 1. A plane triangulation of order 5. The edges ac and bc are braced by de; ad and bd are braced by ce; ae and be are braced by cd; cd, ce and de are unbraced. an edge is unbraced, it can be deleted and the resulting quadrilateral face can have its other diagonal drawn in to produce a new plane triangulation. This is the diagonal Australasian Journal of Combinatorics 2(1990) pp 121-133 operation studied in [1], and shown there to be essentially capable of transforming any plane triangulation of order n into any other. A braced edge may be regarded as an obstruction to diagonal operations, so it is of interest to study the possible number and distribution of braced edges in a plane triangulation. As shown in [1], if T is any plane triangulation of order n:::'5, any edge which braces another must itself be unbraced. (Here we shall refer to this result as Theorem 0.) Thus plane triangulations of order 12:::.5 always contain un braced edges. But a single edge can brace more than one edge, so the possible proportion of braced to unbraced edges is not apparent. We call an un braced edge neutral if it does not brace any other edge. The triangulation in Figure 1 has no neutral edges. The possible number and distribution of neutral edges are also matters of some interest. In section 5 of the paper we show that any fixed number of braced edges can be achieved by triangulations of all sufficiently large orders. In section 7 we establish that the maximum number of braced edges in a triangulation of order n is 2n-4 when n=2 (mod 3) and 2n-5 otherwise, and in Section 8 we describe configurations which achieve these maximum values. Finally, Theorem 4 of Section 9 summarizes our result that for any r less than the maximum (with exactly one exception) there is a triangulation with exactly r braced edges. 2. Triangulations of Small Order Let us begin by exanlining the (equivalence classes of) plane triangulations of small order. This will lay the foundation for our subsequent results. The plane triangulation T(3) of order 3 corresponds to a plane embedding of the complete graph K 3' It is degenerate in that its two faces have the same boundary C, and our definition of braced edges does not apply. However, the plane triangulation T(4) of order 4 can be regarded as a refinement of it, obtained by insertion of a vertex a in the interior of C, the resultant triangulation of the interior being unique. Hence T(4) is the unique plane triangulation of order 4. It corresponds to a plane embedding of the complete graph K4' and all 6 of its edges are braced. Continuing with the triangulation just obtained, we can funher refine it by insertion of a venex b in the exterior of C, the resultant triangulation of the exterior being unique. This produces a plane triangulation T(5) of order 5 in which the 6 edges incident with a or b are all braced, while the 3 edges of Care un braced but none is neutral (Figure 1). It is straightforward to verify that T(5) is the unique plane triangulation of order .s. It is a plane embedding ofKs-e, the complete graph of order 5 with one edge deleted.