: dual graphs Math 104, Graph Theory March 28, 2013 : dual graphs Duality Definition Given a plane graph G, the dual graph G is the plane graph whose vtcs are the faces of G. The correspondence between edges of G and those of G is as follows: if e 2 E(G) lies on the boundaries of faces X and Y, then the endpts of the dual edge e 2 E(G ) are the vtcs x and y that represent faces X and Y of G. : dual graphs
Example of a dual graph A dual graph of a plane graph. : dual graphs Example of a dual graph A dual graph of a plane graph. : dual graphs
Correspondence between G and G vtcs of G =) faces of G edges of G =) edges of G dual of a plane graph =) multigraph loops of G =) cut edges of G multiple/parallel edges of G =) distinct faces of G w/multiple common boundary edges : dual graphs Observations about a dual graph I The dual G of a plane graph G is itself a plane graph. (Intuitively clear that we can draw the dual as a plane graph, but we do not prove this fact.) I The dual of any plane graph is connected. Question Given a planar graph G, do all planar embeddings of G have the same dual (isomorphic duals)? : dual graphs
Observations about a dual graph I The dual G of a plane graph G is itself a plane graph. (Intuitively clear that we can draw the dual as a plane graph, but we do not prove this fact.) I The dual of any plane graph is connected. I Two embeddings of a planar graph may have nonisomorphic duals. I (G ) = G if and only if G is connected. (Requires proof Exercise 6.1.18 of text.) Preview: planar graphs and coloring coloring regions of map! coloring faces of planar embedding! coloring vtcs of G : dual graphs
: characterizations and special classes of planar graphs Math 104, Graph Theory March 28, 2013 Subdivisions Definition The subdivision of an edge e = uv is the replacement of e with new vertex w and two new edges uw and wv (replace edge w/path of length 2). u v u w v Definition A graph H is a subdivision of graph G if one can obtain H from G by a series of edge subdivisions.
Example of a subdivision Petersen graph contains a K 3,3 -subdivision. Example of a subdivision Petersen graph contains a K 3,3 -subdivision.
Minors Definition A minor of a graph G is any graph obtainable from G via a sequence of vertex and edge deletions and edge contractions. Petersen graph has a K 5 -minor. Subdivisions and minors Lemma If G contains a subdivision of H, then H is a minor of G. As an example, let G be the Petersen graph and H be K 3,3. We have already seen that the Petersen graph contains a subdivision of K 3,3. Furthermore, K 3,3 is a minor of the Petersen graph as seen below: blue = delete red = contract Petersen graph has a K 3,3 -minor.
Subdivisions and minors Lemma If G contains a subdivision of H, then H is a minor of G. As an example, let G be the Petersen graph and H be K 3,3. We have already seen that the Petersen graph contains a subdivision of K 3,3. Furthermore, K 3,3 is a minor of the Petersen graph as seen below: K 3,3 blue = delete red = contract Petersen graph has a K 3,3 -minor. Characterizations of planar graphs Theorem (Kuratowski) A graph G is planar if and only if contains no subdivision of K 5 or K 3,3. Kuratowski s Theorem... I provides a structural characterization of planar graphs I is useful in showing that a graph is nonplanar Theorem (Wagner) A graph G is planar if and only neither K 5 nor K 3,3 is a minor of G.
Special kinds of planar graphs Definition A maximal planar graph is a simple planar graph that is not a spanning subgraph of another planar graph. In other words, it is a planar graph to which no new edge can be added without violating the planarity. Definition A triangulation is a simple plane graph in which all faces have degree 3. Maximal planar graphs Proposition For a simple n-vertex plane graph G with n 3, the following are equivalent. 1. G is a maximal plane graph. 2. G is a triangulation. 3. G has 3n 6 edges. Proof. 1 ) 2: Suppose BWOC 9 a face of degree 4 or more in the maximal plane graph G. Then 9 vtcs u and v on this face boundary such that uv 62 E(G); if not, a vertex and incident edges can be added to the interior of the face to obtain a planar embedding of K 5.
Maximal planar graphs Proposition For a simple n-vertex plane graph G with n 3, the following are equivalent. 1. G is a maximal plane graph. 2. G is a triangulation. 3. G has 3n 6 edges. Proof. 1 ) 2: Suppose BWOC 9 a face of degree 4 or more in the maximal plane graph G. Then 9 vtcs u and v on this face boundary such that uv 62 E(G); if not, a vertex and incident edges can be added to the interior of the face to obtain a planar embedding of K 5. Drawing the edge uv in the interior of the face does not violate the planarity of G and hence contradicts the maximality of G. )( Maximal planar graphs Proposition For a simple n-vertex plane graph G with n 3, the following are equivalent. 1. G is a maximal plane graph. 2. G is a triangulation. 3. G has 3n 6 edges. Proof. 2 ) 3: Suppose G is a triangulation with f faces. Then 2 E(G) = Â d(face)=3f faces and so by Euler s formula, 2 = n e + f = n e + 2 3 e = n 1 3 e. Thus, e = 3n 6.
Maximal planar graphs Proposition For a simple n-vertex plane graph G with n 3, the following are equivalent. 1. G is a maximal plane graph. 2. G is a triangulation. 3. G has 3n 6 edges. Proof. 3 ) 1: By previous theorem, E(G) apple 3n 6, so if G has 3n 6 edges, no edge can be added to the planar graph G. Maximal planar graphs Remark Every simple planar graph is a spanning subgraph of a maximal planar graph. Procedure to get maximal planar graph from simple plane graph I In a plane graph G with at least 3 vtcs, if a face has a cut edge inside it, add an edge between one end of cut edge and any other vertex on boundary of face. I Otherwise, a face boundary is a k-cycle with k 3. If k 4, edges (chords) can be drawn from one vertex on the cycle to every other vertex on the cycle so that the face is divided into triangles.
Example of generating a maximal planar graph Example of generating a maximal planar graph Adding edges to obtain a maximal planar graph.
Outerplanar graphs Definition A graph is outerplanar if it has a planar embedding in which all vtcs are on the boundary of the unbounded face. (Equivalently, some face in the embedding includes every vertex on its boundary.) Remarks I All cycles are outerplanar. I All trees are outerplanar. I We can use Kuratowski s Thm to show that G is outerplanar if and only if G contains no subdivision of K 4 or K 2,3. (Exercise 6.2.7 of textbook) I Every simple outerplanar graph has a vertex of degree at most 2, i.e., d(g) apple 2.