Three-dimensional Tutte embedding

this paper.
Theorem A (Steinitz, see Grunbaum 3]) A graph G is 3-polytopal if and only
if G is planar and 3-connected.
Theorem B (Tutte 7]) If G is 3-connected planar graph, then the facial cycles in
Conditions are given for a graph to have a convex representation in three dimensions. This extends Tutte's barycentric embedding. Introduction
Tutte 7] proved that a 3-connected planar graph has a nice embedding in the plane in the sense that it can be drawn so that the boundary of every face is a convex polygon and every edge is a straight line segment. This is called a convex embedding or convex representation. More details on Tutte embedding are given in the next section. The main objective of this work is to visualize nice drawings of nonplanar graphs in three-dimensional space and to introduce some ideas for future research. Now we describe some basic notation to be used throughout this paper. Let G = (V; E ) represent a nite graph with vertex set V and edge set E . A cycle C in G is a collection of distinct vertices v1; v2; : : :; vn such that vi is adjacent to vi+1 for 1 i n, where the indices are considered modulo n. If there are no edges in G between any two nonconsecutive edges of C , then C is chordless. If G can be drawn on the plane so that the edges are Jordan arcs whose end points are vertices and no two edges meet except at the vertices, then G is called a planar graph. Given a plane drawing of a planar graph G, the graph G itself (the vertices and the edges together) can be viewed as a subset of <2. It is convenient to call this subset G. Then <2 ? G is a union of simply connected regions called faces of G, and exactly one of these regions is in nite. The boundary of any face of a 2-connected plane graph corresponds to a cycle of the graph. If the cycle C = v1; v2; : : :; vn corresponds to a face, then we call C a facial cycle . The facial cycle that corresponds to the in nite face is called the outer cycle. Suppose G is a graph and H is a subgraph of G, then we say H is nonseparating if G ? V (H ) (also written G ? H ) is connected. Let Q be a convex polygon in the plane with vertices x1; x2; : : :; xn. We say that the vertices of Q are listed in cyclic order if xixi+1 is a side of Q for 1 i n, where the indices are considered modulo n. An n-wheel is a graph Wn+1 consisting of a cycle v1; v2; : : :; vn and a vertex v adjacent to every vertex of the cycle. This paper presents an approach for the display of certain nonplanar graphs in 3-space. In Section 2, we have a detailed discussion of barycentric embeddings of which Tutte's embedding is a special case. In Section 3, we have drawings of several graphs using NETPAD ( a computer program developed at Bellcore). In Section 4 we consider two di erent barycentric embeddings of G K2 where G is 3-connected and prove that these embeddings are nice in a certain well-de ned sense. In Section 5, we give a negative answer to a question posed by Tutte 7]. The reader is referred to any standard text book on Graph Theory for the unde ned terms. Several of the terms may also be found in Thomassen's article 5].
any plane embedding of G are those cycles which are chordless and nonseparating. tion and any properly chosen convex polygon can play the role of outer cycle.
Three-Dimensional Tutte Embedding
Kiran Chilakamarri Dept. of Math. and Computer Science Central State University Wilberforce, OH 45384 Nathaniel Dean Michael Littman AT&T Bell Laboratories Dept. of Computer Science 600 Mountain Avenue Brown University Murray Hill, NJ 07974 Providence, RI 02912 May 5, 1995
2 The Tutte Embedding
The graph G(P ) of a d-polytope P (i.e., P is the convex hull of a nite set of points in <d ) is the graph consisting of the vertices and the edges of P . If d = 3, then an appropriate projection of P onto the plane provides a convex representation (or a convex drawing, i.e., every face boundary is a convex polygon) of the graph G(P ) in the plane. A graph G is d-polytopal if there is a d-polytope P such that G is isomorphic to G(P ). The following theorems of Steinitz and Tutte are of interest for 2
Theorem C (Tutte 6]) Every 3-connected planar graph has a convex representaThe Tutte embedding or barycentric mapping is de ned as follows. Let G be a 3-connected planar graph, and let J = v1; v2; : : : ; vk v1 be a facial cycle of G. Let C be a convex polygon in the plane with k vertices x1; x2; : : : ; xk listed in the cyclic order. Let f : V (J ) ! <2 be a map de ned by f (vi) = xi for i = 1; 2; : : : ; k. We can extend the map f to all of V (G) as follows: X f (v) = d(1v) f (u): uv2E Tutte 7] shows that this extension is feasible and unique, and it results in an embedding of G in the plane with the property that C is the outer facial cycle, every face in the embedding is convex, and no two edges cross. Notice that the range of the barycentric map could be <d for any d 2 and J could be any subgraph of G embedded in <d . The resulting embedding of course may or may not be interesting (or nice). However the extension of the barycentric map f is possible and unique whenever the subgraph G ? J is connected. Let G be any connected graph with a nonseparating 3-polytopal subgraph H . Then H is isomorphic to H (P ) for some convex polytope P in <3. Let f be a function that maps the vertices of H to the corresponding vertices of P . Then we can extend f to the rest of the vertices of G by f (v) = Puv2E f (u)=d(v). In this paper any such embedding of a graph in a higher dimensional space is called a Tutte embedding or barycentric mapping. In the next section we show embeddings of several graphs. We used the software tools NETPAD 2] and XGobi 1] to construct and view the embeddings in 3-space.