Characterizations of total unimodularity

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Suppose a constraint matrix $A in -1,0,1^m times n$ is totally unimodular. Then we know that every square submatrix consisting of exactly two non-zero entries per row and per column is totally unimodular. My question is: Is the converse true? That is, if every square submatrix of $A$ consisting of exactly two non-zero entries per row and per column is totally unimodular, does that mean $A$ is totally unimodular? (Apparently it seems that the second is a weaker condition, because total unimodularity requires every square submatrix of $A$ to be totally unimodular, and not just the ones consisting of exactly two non-zero entries per row and per column, but I'm not sure.)



If not, is there any other kind of restriction on $A$ that we can put such that this condition becomes sufficient? (For example, two rows (i.e. constraints) can have a maximum of one column in common such that it is non-zero for both rows)




graph-theory total-unimodularity




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    Suppose a constraint matrix $A in -1,0,1^m times n$ is totally unimodular. Then we know that every square submatrix consisting of exactly two non-zero entries per row and per column is totally unimodular. My question is: Is the converse true? That is, if every square submatrix of $A$ consisting of exactly two non-zero entries per row and per column is totally unimodular, does that mean $A$ is totally unimodular? (Apparently it seems that the second is a weaker condition, because total unimodularity requires every square submatrix of $A$ to be totally unimodular, and not just the ones consisting of exactly two non-zero entries per row and per column, but I'm not sure.)



    If not, is there any other kind of restriction on $A$ that we can put such that this condition becomes sufficient? (For example, two rows (i.e. constraints) can have a maximum of one column in common such that it is non-zero for both rows)




    graph-theory total-unimodularity




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      down vote

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      Suppose a constraint matrix $A in -1,0,1^m times n$ is totally unimodular. Then we know that every square submatrix consisting of exactly two non-zero entries per row and per column is totally unimodular. My question is: Is the converse true? That is, if every square submatrix of $A$ consisting of exactly two non-zero entries per row and per column is totally unimodular, does that mean $A$ is totally unimodular? (Apparently it seems that the second is a weaker condition, because total unimodularity requires every square submatrix of $A$ to be totally unimodular, and not just the ones consisting of exactly two non-zero entries per row and per column, but I'm not sure.)



      If not, is there any other kind of restriction on $A$ that we can put such that this condition becomes sufficient? (For example, two rows (i.e. constraints) can have a maximum of one column in common such that it is non-zero for both rows)




      graph-theory total-unimodularity




      share|cite|improve this question











      Suppose a constraint matrix $A in -1,0,1^m times n$ is totally unimodular. Then we know that every square submatrix consisting of exactly two non-zero entries per row and per column is totally unimodular. My question is: Is the converse true? That is, if every square submatrix of $A$ consisting of exactly two non-zero entries per row and per column is totally unimodular, does that mean $A$ is totally unimodular? (Apparently it seems that the second is a weaker condition, because total unimodularity requires every square submatrix of $A$ to be totally unimodular, and not just the ones consisting of exactly two non-zero entries per row and per column, but I'm not sure.)



      If not, is there any other kind of restriction on $A$ that we can put such that this condition becomes sufficient? (For example, two rows (i.e. constraints) can have a maximum of one column in common such that it is non-zero for both rows)





      graph-theory total-unimodularity





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