graph

Question

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

Answer 1

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.

Comments

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.