By Christopher Griffin

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6(b) represents the communications connections between individuals in two terrorist cells. 6. We illustrate a vertex cut and a cut vertex (a singleton vertex cut) and an edge cut and a cut edge (a singleton edge cut). Cuts are sets of vertices or edges whose removal from a graph creates a new graph with more components than the original graph. two individuals who communicate between these two cells could be important for finding a way to capture or disrupt this small terrorist network. 36. Let G = (V, E) be a connected graph and let e ∈ E.

Illustration of the main argument in the proof that a graph is bipartite if and only if all cycles have even length. 7) U2 = V2k k=1 Clearly there can be no edge connecting a vertex in U1 with a vertex in U2 . This completes the proof. 6. 45 (Acyclic Graph). A graph that contains no cycles is called acyclic. 46 (Forests and Trees). Let G = (V, E) be an acyclic graph. If G has more than one component, then G is called a forest. If G has one component, then G is called a tree. 47. 10. Note that a tree (if drawn upside down) can be made to look exactly like a real tree growing up from the ground.

Um and edges f1 , . . , fm so that: c = (u1 , f1 , . . , um , fm , u1 ) is a cycle and e is among the f1 , . . , fm . Without loss of generality, assume that e = fm and that e = {um , u1 }. ) Then in G we will have the path: c = (u1 , f1 , . . , um ) The fact that e is in the walk w implies there are vertices vi and vi+1 so that e = {vi , vi+1 } (with vi = u1 and vi+1 = um ). In deleting e from G we remove the sub-walk (vi , e, vi+1 ) from w. But we can create a new walk with structure: w = (v1 , e1 , .

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Graph and Network Algorithms [Lecture notes] by Christopher Griffin

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