# graph to tree and graph isomorphism problem

Sorry if the following are stupid questions (i do not know much about the graph theory).

1. Motivation

we do not know the graph isomorphism problem in class P or NP complete and it is P in the case trees (see http://en.wikipedia.org/wiki/Graph_isomorphism_problem).

2. Transformation a graph to tree

Let $G$ be a simple connect graph with $v$ vertices and $e$ edges. We consider the following process:

Process: If $G$ is not a tree we will have a cycle in $G$, called $C_k$ with vertices (assume that) $v_1,...,v_k$. Then

• We add a new vertex $w$ to $G$
• delete $k$ edges of $C_k$: $(v_1,v_2),...,(v_k, v_1)$.
• Add $k$ edges $(w,v_1),...,(w,v_k)$ to the graph.

We obtain a new graph $G'$ with $v+1$ vertices and $e$ edges.

Theorem. Applying the above process $v - e + 1$ times we get a tree $T$.

Notice that we may get many trees by different processes.

Question 1. Assume that we always choice $C_k$ such that $k$ as small as possible. Is it true that every tree we obtain are isomorphism?

I do not know this question true or false. In the case we have a negative in general, which condition it is true?

3. The graph isomorphism problem

In the case Question 1 has an affirmative answer for each $G$ we get a tree $T(G)$ (up to an isomorphism).

Question 2. Assume that $G$ and $H$ are simple connect graph with same number of vertices, edges, degrees of vretex,... (we can need more condition which easy to check). Whether $G$ and $H$ are isomorphism iff $T(G)$ and $T(H)$ are isomorphism?

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If two cycles of minimum length have a common edge, then it matters which is chosen. Try two triangles with a common edge, plus one more vertex joined to an apex of one of the triangles.

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we choice one of them and get a new graph. –  Pham Hung Quy Jul 4 '12 at 9:55
one process we choice one cycle. –  Pham Hung Quy Jul 4 '12 at 9:57
If you try the different possible processes with the graph suggested by Brendan, you will see that the answer to question 1 is no. –  nvcleemp Jul 4 '12 at 10:20
Thanks you a lost! –  Pham Hung Quy Jul 4 '12 at 13:52