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16 is a partial subgraph of the graph of Fig. 1-13. Reduction of a graph. Let G = (X; I') be a graph; define over X a partition X i , X2, .. , r. , Xr} be the set of parts thus defined, and let r° over X° be a function defined by: (X, X;) E r° if and only if there exist a point x E Xl and a point y e Xj such that (x, y) E r. The graph G° = (X°, r°) is a reduced graph of the graph G. 17 presents a graph G and a reduced graph G° of G. Obviously, many reduced graphs can be obtained from a graph. The interest in such an operation is contingent upon the properties of the partition of the points of the graph.

11. A strongly connected graph is semi-strongly connected. A semi-strongly connected graph is quasi-strongly connected. A quasi-strongly connected graph is weakly connected. Proof. Strong connectivity entails semi-strong connectivity: this is selfevident. Semi-strong connectivity entails quasi-strong connectivity: if the path y(xy) exists, it is sufficient to put x = z' and y = z. Quasi-strong connectivity entails weak connectivity: we have, for instance, the chain (xy) consisting of the paths y(xz) and y(yz).

10 Let 0(xy) = (X'; V') be a partial subgraph of G = (X; V) constructed in the following way: X' C X and V' C V; z e X' p e(xz) + e(zy) = e(xy); (z, z') e V' p z and z' e X' and e(xz) + 1 = e(xz'). All the paths y(xy) in 0(xy) are tracks O(xy) in G, and all the tracks 6(xy) of G appear in 0(xy). This theorem follows directly from the two previous ones. It provides an algorithm permitting us to find the tracks 6(xy) in a graph G. Example. Consider the graph G of Fig. 21. We want to construct 0(ac).

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Applications of graph theory to group structure (Prentice-Hall series in mathematical analysis of social behavior) by Claude Flament

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