Combinatorial optimization : theory and algorithms by B H Korte; Jens Vygen

By B H Korte; Jens Vygen

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Just storing the edges in an arbitrary order is not very convenient. Almost all graph algorithms require finding the edges incident to a given vertex. Thus one should have a list of incident edges for each vertex. In case of directed graphs, two lists, one for entering edges and one for leaving edges, are appropriate. This data structure is called adjacency list; it is the most customary one for graphs. For direct access to the list(s) of each vertex we have pointers to the heads of all lists; these can be stored with O(n log m) additional bits.

Moreover, if G is 2-connected, then so is G + P, where P is an x-y-path, x, y ∈ V (G) and x = y: 30 2. Graphs deleting any vertex does not destroy connectivity. We conclude that a graph with a proper ear-decomposition is 2-connected. To show the converse, let G be a 2-connected graph. Let G be the maximal simple subgraph of G; evidently G is also 2-connected. e. it contains a circuit. Since it is simple, G , and thus G, contains a circuit of length at least three. So let H be a maximal subgraph of G that has a proper ear-decomposition; H exists by the above consideration.

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). The strongly connected components of a digraph are the maximal strongly connected subgraphs. 7. In a digraph G, each edge belongs either to a (directed) circuit or to a directed cut. Moreover the following statements are equivalent: (a) G is strongly connected.

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