Hierholzer’s algorithm. Hierholzer’s algorithm is anotheralgorithm for finding a...

Hierholzer’s algorithm. Hierholzer’s algorithm is anotheralgorithm for finding an Euler circuit in a graph. The basicidea behind Hierholzer’s algorithm is to start with an arbitrary circuit and then enlarge it by patching to it a kissingcircuit, continuing this way and making larger and largercircuits until the circuit cannot be enlarged any farther. (Forthe definition of kissing circuits, see Exercise 74.) More formally, Hierholzer’s algorithm is as follows:

Step 1. Start with an arbitrary circuit C₀.

Step 2. Find a kissing circuit to C₀. If there are no kissingcircuits to C₀, then you are finished—C₀ is itself an Eulercircuit of the graph [see Exercise 74(b)]. If there is a kissingcircuit to C₀, let’s call it K₀, and let V denote the vertex atwhich the two circuits kiss. Go to Step 3.

Step 3. Let C₁ denote the circuit obtained by “patching”K₀ to C₀ at vertex V (i.e., start at V, travel along C₀ backto V, and then travel along K₀ back again to V). Now finda kissing circuit to C₁. (If there are no kissing circuits to C₁,then you are finished—C₁ is your Euler circuit.) If there isa kissing circuit to C₁, let’s call it K₁, and let W denote thevertex at which the two circuits kiss. Go to Step 4.

Steps 4, 5, and so on. Continue this way until there are nomore kissing circuits available.

(a) Use Hierholzer’s algorithm to find an Euler circuit forthe graph shown in Fig. 5-65 (this is the graph modelfor the mail carrier in Example 5.14).

(b) Describe a modification of Hierholzer’s algorithm thatallows you to find an Euler path in a connected graphhaving exactly two vertices of odd degree. (Hint: Apath can also have a kissing circuit.)