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## 3Circuit chain and Petersen minor

### 3.1 Weight decomposition and removable circuit

Definition 3.1.1 Let (G, w) be an eulerian weighted graph. A set of eulerian weighted subgraphs {(Gi, wi) : i = 1, …, t} is called an eulerian weight decomposition of (G, w) if each Gi is a subgraph of G and for every eE(G) (where wi(e) = 0 if eE(Gi)). (See Figure 3.1 where t = 2.)

Definition 3.1.2 Let (G, w) be an admissible eulerian weighted graph. An eulerian weight decomposition {(Gi, wi) : i = 1, …, t} of (G, w) is admissible if each (Gi, wi) is admissible. (See Figure 3.1.)

Definition 3.1.3 Let (G, w) be an admissible eulerian weighted graph. If (G, w ...

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