Show that if every circuit not passing through any vertex

Chapter , Problem 30E

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Show that if every circuit not passing through any vertex other than its initial vertex more than once in a connected graph contains an odd number of edges, then this graph must be a cactus.A degree-constrained spanning tree of a simple graph G is a spanning tree with the property that the degree of a vertex in this tree cannot exceed some specified bound. Degree- constrained spanning trees are useful in models of transportation systems where the number of roads at an intersection is limited, models of communications networks where the number of links entering a node is limited, and so on.In Exercises 31-33 find a degree-constrained spanning tree of the given graph where each vertex has degree less than or equal to 3. or show that such a spanning tree does not exist.