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May 28th, 2008, 04:38 AM  #1 
Newbie Joined: Apr 2008 Posts: 20 Thanks: 0  Why can't a vertex with degree = 1 be cutvertex in a tree?
I read in a book that, every vertex in a tree with degree > 1 is a cutvertex. I wonder why degree > 1 condition should be there. Even a vertex which has degree = 1 can also be a cutvertex. Isn't it? So, could someone please explain whether I am right or that degree > 1 condition is really required. Just to make sure that there is no ambiguity in the discussion, I am adding the definitions from the book itself. The vertex connectivity of a connected graph G is defined as the minimum number of vertices whose removal from G leaves the remaining graph disconnected. A connected graph is separable if its vertex connectivity is 1. In a separable graph a vertex whose removal disconnects the graph is called a cutvertex. 
May 28th, 2008, 07:30 AM  #2 
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3 
In order for a vertex to connect two different subgraphs, it needs an edge connected to each subgraph, thus it needs degree two. As a littlle "experiment" to help you grasp the concept, draw 2 (or 3 or 4 or...) graphs next to each other, and try to connect any number of them with 1 vertex. How many edges do you need? Now, try to connect any two of them with 1 edge. Can you do it using an extra vertex? The thing I don't understand is "[b]any[b] vertex with degree > 1 is a cut vertex", but it may have to do with the type of graph you're working with. 

Tags 
cutvertex, degree, tree, vertex 
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