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Chapter 3: Q27E (page 111)

Two paths in a graph are called edge-disjointif they have no edges in common. Show that in any undirected graph, it is possible to pair up the vertices of odd degree and find paths between each such pair so that all these paths are edge-disjoint.

Short Answer

Expert verified

In any undirected graph. It is possible to pair up the vertices of odd degrees and the paths between each pair can be found so that all these paths are edge-disjoint.

Step by step solution

01

Explain edge-disjoint

In a graph, if there is no common edge that exists between the two paths, then it is called edge-disjoint.

02

Step 2: Show that it is possible to pair up the vertices of odd degree and find paths between each such pair so that all these paths are edge-disjoint.

Consider any undirected graph that has multiple edges and vertices. In an undirected graph, the number of vertices with an odd degree must be even and it can be divided into two groups.

In all connected branches of the graph, the number of odd-degree vertices is even n . For every odd-degree vertices, there must be a pair of connected vertices (u,v) and the path edge is removed from the graph. Consider a graph containing n-1 pairs of odd-degree vertices, and repeat the process to find each pair. The paths between vertices are disjoint.

Therefore, it has been proved that It is possible to pair up the vertices of odd degrees and the paths between each pair can be found so that all these paths are edge-disjoint.

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Most popular questions from this chapter

Run the DFS-based topological ordering algorithm on the following graph. Whenever you have a choice of vertices to explore, always pick the one that is alphabetically first.

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