By Qifan Y.

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**Extra resources for A 2. 79 competitive online algorithm for two processor real-time systems with uniform value density**

**Example text**

Then there exists an edge e such that G/e is also 3-connected. Proof: Suppose there is no such edge. e. has a connected component C with |V (C)| < |V (G)| − 3. Choose e, x and C such that |V (C)| is minimum. x has a neighbour y in C, because otherwise C is a connected component of G − {v, w} (but G is 3-connected). e. there exists a vertex z such that G − {x, y, z} is disconnected. Since {v, w} ∈ E(G), there exists a connected component D of G − {x, y, z} which contains neither v nor w. But D contains a neighbour d of y, since otherwise D is a connected component of G − {x, z} (again contradicting the fact that G is 3-connected).

Y form an undirected circuit with the properties (a). Case 2: x has not been labelled. Then let R consist of all labelled vertices. Obviously, the undirected cut δ + (R) ∪ δ − (R) has the properties (b). Suppose that an undirected circuit C as in (a) and an undirected cut δ + (X ) ∪ − δ (X ) as in (b) both exist. All edges in their (nonempty) intersection are black, they all have the same orientation with respect to C, and they all leave X or all enter X . This is a contradiction. A digraph is called strongly connected if there is a path from s to t and a path from t to s for all s, t ∈ V (G).

On the other hand, because of the minimality of F, F \ {{v, w}} is not an odd cover (for any w with {v, w} ∈ F), so |δG (V (Ci ))| is odd for i = 1, . . , k. Since k |δG (V (Ci ))| = |δG (X )| + |δG (v)| i=1 − 2|E G ({v}, V (G) \ X )| + 2 |E G (Ci , C j )|, 1≤i< j≤k we conclude that k has the same parity as |δG (v)|. 2. A bipartition of an undirected graph G is a partition of the vertex set V (G) = . A ∪ B such that the subgraphs induced by A and B are both empty. A graph is called . bipartite if it has a bipartition.