Planar Graphs In the first half of this book, we consider mostly planar graphs and their ge- ometric representations, mostly in the plane. We start with a survey of basic results on planar graphs. This chapter is of an introductory nature; we give very few proofs. Planar graphs play a very important role, and some of the considerations here provide a good introduction into the methods in this book, so this chapter deserves to be part of the main material. We assume familiarity with basic graph theory. 1 Graphs and maps 1.1 Planarity and duality A multigraph G is planar, if it can be drawn in the plane so that its edges are Jordan curves and they are disjoint except for their endnodes. A planar map is a planar multigraph with a fixed embedding. We also use this phrase to denote the image of this embedding, i.e., the subset of the plane which is the union of the set of points representing the nodes and the Jordan curves representing the edges. The complement of a map G decomposes into a finite number of arcwise connected pieces, which we call the countries of the planar map. (Often the countries are called \faces", but we reserve the word face for the faces of poly- hedra, and the word facet for maximum dimensional proper faces of polyhedra.) The set of its countries will be denoted V ∗, and often we use the notation f = jV ∗j. There is one special country, which is unbounded; the other countries are bounded. A planar map is called a triangulation if every country has 3 edges. Note that a triangulation may have parallel edges, but no two parallel edges can bound a country. In every simple planar map G we can introduce new edges to turn all countries into triangles while keeping the graph simple. If the multigraph is connected (which we are going to assume most of the time), then every country is homeomorphic to an open disc, except for the unbounded country, which is homeomorphic to an open ring. Each country has a boundary consisting of a cyclic sequence of edges. An edge can occur twice in the boundary sequence; this happens if and only if it is a cut-edge (isthmus) of the multigraph. If the multigraph is 2-edge-connected, then no boundary sequence contains a repetition, and if it is 2-node-connected, then every boundary is a (simple) cycle. The country also defines a cyclic sequence of nodes; a node my occur many times in this sequence. Each occurrence of a node in this sequence is called a corner. In other words, a corner is defined by two edges of the boundary of the country, called the edges of the corner, which are consecutive in the cyclic order along the boundary, but also in the cyclic order of the edges around the node. These two edges are different except if the corner is a node of degree 1. 1 Every connected planar map G has a dual map G∗ = (V ∗;E∗) (Figure 1). As an abstract graph, this can be defined as the multigraph whose nodes are the countries of G, and if two countries share k edges, then we connect them in G∗ by k edges. So each edge e 2 E will correspond to a unique edge e∗ of G∗, and vice versa; hence jE∗j = jEj = m. If the same country is incident with e from both sides, then e∗ is a loop. For a country p, we denote by deg(p) its degree in the dual map. This is the number of edges on its boundary, with the addition that if the boundary passes an edge twice, then this edge is counted twice in deg(v). This dual map has a natural drawing in the plane: in the interior of each country F of G we select a point vF (which can be called its capital), and on ∗ each edge e 2 E we select a point ue (this will not be a node of G , just an auxiliary point). We connect vF to the points ue for each edge on the boundary of F by nonintersecting Jordan curves inside F . If the boundary of F goes through e twice (i.e., both sides of e belong to F ), then we connect vF to ue by two curves, entering e from two sides. The two curves entering ue form a single Jordan curve representing the edge e∗. It is not hard to see that each country of G∗ will contain a unique node of G, and so (G∗)∗ = G. Figure 1: A planar map and its dual. Often we will need an orientation of G. Then each (oriented) edge e 2 E has a tail t(e) 2 V , a head h(e) 2 V , a right shore r(e) 2 V ∗, and a left shore l(e) 2 V ∗. There is a natural way to define an orientation of the dual map G∗ = (V ∗;E∗), so that the dual edge e∗ 2 E∗ crosses the corresponding edge e 2 E from left to right. The dual of the dual of an oriented map is not the original map, but the original map with its orientation reversed. From a planar map, we can create further planar maps that are often useful. The lozenge map G♦ = (V ♦;E♦) of a planar map G is defined by V ♦ = V [ V ∗, where we connect p 2 V to i 2 V ∗ if p is a node of country i. So G♦ has an edge for every corner of G. We do not connect two nodes in V nor in V ∗, so G♦ is a bipartite graph. It is easy to see that G♦ is planar. The dual map of the lozenge map is also interesting. It is called the medial map of G, denoted by G./, and it can be described as follows: we subdivide each edge by a new node; these will be the nodes of the medial map. For every 2 corner of every country, we connect the two nodes on the two edges bordering this corner by a Jordan curve inside the country (Figure 2). The medial graph has two kinds of countries: those corresponding to nodes of G and those corre- sponding to countries of G. Thus the countries of G./ can be two-colored so that every edge separates countries of different colors. This of course is equivalent to saying that G♦ is bipartite. Figure 2: The dual map, the lozenge map and the medial map of a part of a planar map. We can avoid the complications of the exceptional unbounded country if we consider maps on the sphere. Every planar map can be projected onto the sphere by inverse stereographic projection (Figure 3), and vice versa. Often this leads to simpler statements, since one does not need to distinguish an unbounded country. (On the other hand, it is easier to follow arguments in the plane.) Figure 3: Projecting a planar map onto the sphere by inverse stereographic projection. 1.2 Euler's Formula We often need the following basic identity for connected planar maps, called Euler's Formula (where n denotes the number of nodes, m denotes the number of edges, and f denotes the number of countries): n − m + f = 2: (1) 3 This follows easily from the observation that if we take a spanning tree of a planar map, then those edges of the dual graph which correspond to edges outside this tree form a spanning tree of the dual graph (Figure 4). The spanning tree has n − 1 edges, the spanning tree of the dual has f − 1 edges, whence (n − 1) + (f − 1) = m. Figure 4: Spanning trees in two planar graphs, and the corresponding spanning trees in their duals. For the grid on the right, the broken edges emenating through the boundary go to the capital of the infinite country, which is not shown. Let us mention some important consequences of Euler's Formula. For a simple planar map, each country has at least 3 edges, so we have 2m ≥ 3f = 3(m + 2 − n), and rearranging, we get m ≤ 3n − 6: (2) If, in addition, the graph has no triangles, then a similar simple counting argu- ment gives that m ≤ 2n − 4: (3) From these inequalities it is easy to derive the following fact. Proposition 1 Every simple planar graph has a node with degree at most 5, and every triangle-free simple planar graph has a node with degree at most 3. From (2) and (3) it follows immediately that the \Kuratowski graphs" K5 and K3;3 (see Figure 5) are not planar. This observation leads to the following characterization of planar graphs. Theorem 2 (Kuratowski's Theorem) A graph G is embeddable in the plane if and only if it does not contain a subgraph homeomorphic to the complete graph K5 or the complete bipartite graph K3;3. To show a further typical application of Euler's Formula, we prove a lemma, which will be useful in several arguments later on. In a planar map whose edges are 2-colored, we call a node quiet if the edges of each color emanating from it are consecutive in the cyclic order of the embedding. 4 KK K 5 3,3 5 K 3,3 Figure 5: The two Kuratowski graphs. The two drawings on the right show that both graphs can be drawn in the plane with a single crossing. Lemma 3 In every simple planar map whose edges are 2-colored there are at least two quiet nodes.