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By Xu B.G.

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Additional resources for A 3-color Theorem on Plane Graphs without 5-circuits

Sample text

Looking at the path relationships in a network naturally partitions a network into different connected subgraphs that are commonly referred to as components. Again, definitions are first provided for undirected networks and later for directed ones. A network (N, g) is connected (or path-connected) if every two nodes in the network are connected by some path in the network. That is, (N, g) is connected if for each i ∈ N and j ∈ N there exists a path in (N, g) between i and j . A component of a network (N, g) is a nonempty subnetwork (N , g ) such that ∅ = N ⊂ N, g ⊂ g, (N , g ) is connected, and if i ∈ N and ij ∈ g, then j ∈ N and ij ∈ g .

I alternate between the different representations as is convenient. It will also be useful to write g ⊂ g, to indicate that {ij : ij ∈ g } ⊂ {ij : ij ∈ g}. Let the shorthand notation of g + ij represent the network obtained by adding the link ij to an existing network g, and, let g − ij represent the network obtained by deleting the link ij from the network g. We can represent directed networks in an analogous manner, viewing ij as a directed link and distinguishing between ij and j i. Let G(N ) be the set of all undirected and unweighted networks on N .

The definitions depend on whether we want to keep track of the direction of the links, and in various applications depend on whether communication is restricted to following the direction of the links or can move in both directions along a directed link, as for example in a network of links between web pages. In the case in which direction is important, the definitions are just as stated above for undirected networks, but the ordering of the nodes in each link now takes on an important role. For instance, we might be interested in knowing whether one web page can be found from another by following (directed) links starting from one page and leading to the other.

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