Signing of a binary matrix to a totally unimodular matrix

Question

I have the following binary matrix: \begin{pmatrix} 1& 1& 1& 0 \\ 0& 1& 1& 1\\ 1& 0& 1& 1\\ 1& 1& 0& 1\\ \end{pmatrix}

Definition: Signing a matrix means means changing some $1$s to $-1$s.

Question. Is there a signing of the matrix such that it becomes totally unimodular over $\Bbb R$?

Answer 1

For example $$ \left[ \begin {array}{cccc} 1&1&1&0\\ 0&-1&-1&1 \\ -1&0&-1&-1\\ -1&-1&0&1 \end {array} \right] $$ (and many others)

EDIT: For the new question, the answer is no. There are only $2^{12}$ possibilities, and it's easy to write a program to check all of them.

Answer 2

This is a four element matrix. All four element matrices except for one (U2,4) are regular so there will be a signing that makes it totally unimodular. Maybe you are asking a more general question? "Given any binary matrix is there a signing that will make it totally unimodular." In this case better to ask whether or not a given binary matrix is regular. Tutte gave a good algorithm for determining if a binary matrix is graphic. Together with Seymour's decomposition of binary matrices we can determine if a binary matrix is regular.