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Which of the following statements is/are TRUE?
[P] Every disconnected graph has an isolated vertex
[Q] A graph is connected if and only if some vertex is connected to all other vertices
[R] The edge set of every closed trail can be partitioned into edge sets of cycles
[S] If a maximal trail in a graph is not closed, then its endpoints have odd degree
asked in Graph Theory by Active (2.1k points)   | 124 views

1 Answer

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P)False. Disconnected means not connected. There may or may not be isolated vertex.

Q)False. In a cycle no vertex connected to all other vertices.

R)False. Partition are disjoint subset of a set. That may or may not a cycle.

S)True.If Trail or Walk is not closed, endpoint must contain some odd degree.
answered by Veteran (54.9k points)  
Q) is not true. It is only true for complete graph.

R)is not true. U cannot consider only a perticular graph. General case u have to assume.

If u are confirmed with ur answer give me some pic what u want to tell

i din't get the solution


any conclusion?
Even S isnt true. Take a cycle graph with 4 vertices i.e. square ABCD and connect BD. Now, trace the max trail as ABCDB but A doesnt have odd degree.

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