A proof of Tutte's realizability condition

Abstract

This paper gives a simple proof of Tutte's realizability condition for a cutset (circuit) matrix of a nonoriented graph [1],[2]. First, a minimum nonrealizable matrix is defined as~a matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[ N U]</tex> that satisfies 1) <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[N U]</tex> is not a cutset (circuit) matrix, 2) <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[ N U]</tex> does not satisfy the conditions in Tutte's theorem, and 3) deleting any column of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N1</tex> or any row of any normal form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[N1 U]</tex> of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[N U]</tex> , the resultant matrix is realizable as a cutset (circuit) matrix. A proof of Tutte's theorem in this paper is accomplished by showing that minimum nonrealizable matrices do not exist.