Regularized motion analysis Interactive demonstration of several techniques for regularized motion analysis (dip6ex17.02) Iterative inverse filtering Interactive demonstration of iterative inverse filtering; generation of test images with motion blur and defocusing (dip6ex17.03). Elementary morphological operators Interactive demonstration of elementary morphological operators, such as erosion, dilation, opening, and closing (dip6ex18.01)

Commutativity of morphological operators Check if morphological erosion and dilation operators are commutative and prove your conclusion! (Hint: If one of the operators is not commutative, present a counter example.) Hit-miss operator Interactive demonstration of the hit-miss operator (dip6ex18.02) Morphological boundary detection Interactive demonstration of morphological boundary detection (dip6ex18.03)

Morphological operations with gray-scale images Interactive demonstration of morphological operators with gray-scale images (dip6ex18.04) Opening and closing Opening and closing are two of the most important morphological operators. 1. What happens if you apply an opening or a closing with the same structure element several times? 2. What is the structure element for an opening

The Pr¨ufer code  : T → W defined by equations (1.3) and (1.4) is a bisection. Figure 1.6 shows some trees and their Pr¨ufer codes for n = 6 (one for each isomorphism class, see Exercise 4.1.6). Let G be a connected multi graph having exactly 2k vertices of odd degree (k 0). Then the edge set

Give a formula for the degree of a vertex of L(G) (using the degrees in G). In which cases is L() an SRG? Let G be a connected graph. Find a necessary and sufficient condition for L(G) to be Eulerian. Conclude that the line graph of an Eulerian graph is likewise Eulerian, and show that

Determine the minimal number of edges a graph G with six vertices must have if [G] is the complete graph . If G is Eulerian, then L(G) is Hamiltonian. Does the converse hold? Every chessboard of size m × n (where m ≤ n) admits a knight’s cycle, with the following three exceptions: (a) m and

Show that knight’s cycles are impossible for the cases (a) and (b) in Theorem 1.4.7. (Case (c) is more difficult.) Hint: For case (a) use the ordinary coloring of a chessboard with black and white squares; for (b) use the same coloring as well as another appropriate coloring (say, in red and green squares) and

By Corollary 1.5.4, the complete graph  is not planar, as a planar graph on five vertices can have at most nine edges. The complete bipartite graph  has girth 4; this graph is not planar by Theorem 1.5.3, as it has more than eight edges. Show that the graphs which arise by omitting one edge e from either  or  are

Show that the Petersen graph is isomorphic to the complement of the triangular graph . The isomorphisms of a graph G to itself are called automorphisms; clearly, they form a group, the automorphism group of G. In this book we will not study automorphisms of graphs, except for some comments on Cayley graphs in Chapter 9;

Show that the automorphism group of the Petersen graph contains a subgroup isomorphic to the symmetric group S5. Hint: Use Exercise 1.5.11. What is the minimal number of edges which have to be removed from  to get a planar graph? For each n, construct a planar graph having as many edges as possible.