Eulerizing Graphs

Class Notes: Eulerizing and Semi-Eulerizing GraphsRemember:Definition: AnEuler path is a path that passes through every edge of a graph. Not allgraphs have Euler path. The one above does not, for instance, because you can’t getto both vertex G and vertex H in one path.An Euler circuit is a circuit that passes through every edge of a graph.Euler’s Circuit Theorem If a graph is connected and every vertex is even, then it has an Euler circuit.Consequence:If a graph has any odd vertices, then it does not have anEuler circuit.Euler’s Path TheoremIf a graph is connected and has exactly two odd vertices, then it has an Euler path.Any such path must start at one of the odd vertices and end at the other one.Consequence: If a graph has more than two odd vertices, then it cannot havean Euler path.Q: What if we need to have an Euler circuit but can’t because there are too manyvertices of odd degree?A: Add duplicate edges to the graph to make it possible to get an Euler circuit. This iscalled eulerizing a graph.Definition: Take a graph and add duplicate edges to it to make all the vertices even.Then the new graph you obtain is an eulerization of the original graph.Examples1

Example of reason to Eulerize: security guard walking the neighborhood must coverevery street block in his rounds. He wants the route with the fewest redundancies(blocks he repeats, also called deadhead blocks) that will bring him back to where hestarted.Example- 3x3 gridgraph:eulerization: now he can cover all the blocks with only 4deadheads.Important note: An eulerization should not add new vertex adjacencies. It alwaysdoubles edges.Define an OPTIMAL EULERIZATION as an eulerization that adds the fewest possibleedges.2

Example:because we can’t just connect C to B.add lots of edgesSometimes it’s simpler, even preferred, to have an Euler path in a graph instead of anEuler circuit. Just as to make the circuits, we can make an Euler path possible byadding duplicate edges, and will call this process semi-eulerization. One keyexample is in planning a parade route. We want to cover every block in our 3 by 3grid but don’t need, in fact don’t want, the parade to start and end at the same place.graph:semi-eulerize1:this semi-eulerization though has the disadvantage of the starting and ending vertex ofthe path are right next to each other. a parade is usually better off if the starting pointand ending point are far apart. How could we semi-eulerize to keep that in mind?3

try this:(start at second vertex from topleft, end at lower-right corner).notice it added 5 edges so there are more deadhead blocks but the advantage is thestarting point is far from the ending point.we can also semi-eulerize so as to get a path that starts and ends exactly where wedecide.Example: In the graph below, semi-eulerize to get an euler path starting at vertex 1and ending at vertex 5: give the path as well.we just need to leave only vertex 1 and vertex 5 odd. Since they are already odd, thisonly required adding one edge.But what if instead we want an Euler path starting at vertex 1 and ending at vertex 6?4

Add to this the additional factor of "costs" for edges. Here is an example from anothertextbook.A photographer needs to take photos of each of the 11 bridges in River Country. Apicture of the region is shown Every time he has to cross a bridge, it costs thephotographer $25 in tolls. Find an optimal route (ie, minimum cost) that crosses everybridge if...a.) The photographer will start and end her trip in the same place.b.) The photographer has the freedom to choose any starting and ending place for thetripc.) The photographer must start her trip on island B and end it on the Left Bank.5