Is it possible to find the criminal with graph-theoretic methods? I've been presented to a problem: 


  Someone commited a crime. When interrogated, the people, named $G,m,M,J,D$ argued:
  
  
  $G:$ It wasn't $D$; It was $M$.
  $m:$ It wasn't $M$; It wasn't $D$
  $M:$ It was $D$; It wasn't $G$.
  $J:$ It was $M$; It was $m$.
  $D:$ It was $J$; It wasn't $G$.
  
  
  Knowing that each one of them told a lie, who was the criminal?


I'm pretty sure that graph theoretic methods could be used to solve this problem. I tried to write a graph matrix with $a_{ij}=-1$ if it is said that wasn't him and $a_{ij}=1$ if someone said it was him and $0$ otherwise (I left the $0$ entries null for simplicity), then:

$$\begin{vmatrix}
{}&amp;{G}&amp;{m}&amp;{M}&amp;{J}&amp;{D}\\ 
{G}&amp;{}&amp;{1}&amp;{}&amp;{}&amp;{-1}\\ 
{m}&amp;{}&amp;{}&amp;{-1}&amp;{}&amp;{-1}\\ 
{M}&amp;{-1}&amp;{}&amp;{}&amp;{}&amp;{1}\\ 
{J}&amp;{}&amp;{1}&amp;{1}&amp;{}&amp;{}\\ 
{D}&amp;{-1}&amp;{}&amp;{}&amp;{1}&amp;{}
\end{vmatrix}$$

But I'm unable to find some kind of pattern to solve this problem with this matrix. So, Is there something that could be used to find the criminous with graph-theoretic methods?

Besides this, I've tried to think about matrix permutations assuming that the first proposition of each one is a lie and the second is truth, then the first is truth, then the second is a lie, for example:

$$\begin{vmatrix}
{}&amp;{G}&amp;{m}&amp;{M}&amp;{J}&amp;{D}\\ 
{G}&amp;{}&amp;{1_L}&amp;{}&amp;{}&amp;{-1_T}\\ 
{m}&amp;{}&amp;{}&amp;{-1_L}&amp;{}&amp;{-1}\\ 
{M}&amp;{-1_L}&amp;{}&amp;{}&amp;{}&amp;{1}\\ 
{J}&amp;{}&amp;{1_L}&amp;{1}&amp;{}&amp;{}\\ 
{D}&amp;{-1_L}&amp;{}&amp;{}&amp;{1}&amp;{}
\end{vmatrix}$$

And then permute the $L$ies and $T$ruths to obtain a set of matrices and then find some property on one of these matrices that could indicate who's the criminous. But I also can't see some pattern here.
The problem has no solution. One of the statements It was M and It wasn't M must be true, so It wasn't D must be false. Therefore $D$ is guilty. But then both of $M$'s statements are true.
I didn't use any of this matrix stuff. I just simply assumed every statement they made was a lie and for G I constructed the set {D,M'} where M' means M didn't do it. Then for m I have (M, D}. Continuing I have {D', G}, {M', m'}, and {J', G} and conclude that G is the one none of them said did do it therefore G did it.