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In deriving their characterization of the perfect matchings polytope,Edmonds, Lov\'asz, and Pulleyblank introduced the so-called {\em Tight CutLemma} as the most challenging aspect of their work. The Tight Cut Lemma infact claims {\em bricks} as the fundamental building blocks that constitute agraph in studying the matching polytope and can be referred to as a key resultin this field. Even though the Tight Cut Lemma is a matching\textup{(}$1$-matching\textup{)} theoretic statement that consists of purelygraph theoretic concepts, the known proofs either employ a linear programmingargument or are established upon results regarding a substantially wider notionthan matchings. This paper presents a new proof of the Tight Cut Lemma, whichattains both of the two reasonable features for the first time, namely, being{\em purely graph theoretic} as well as {\em purely matching theory closed}.Our proof uses, as the only preliminary result, the canonical decompositionrecently introduced by Kita. By further developing this canonicaldecomposition, we acquire a new device of {\em towers} to analyze the structureof bricks, and thus prove the Tight Cut Lemma. We believe that our new proof ofthe Tight Cut Lemma provides a highly versatile example of how to handlebricks.

A Graph Theoretic Proof of the Tight Cut Lemma