Digraphs: Theory Algorithms and Applications by J??rgen Bang-Jensen;Gregory Z. Gutin

By J??rgen Bang-Jensen;Gregory Z. Gutin

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Conversely, several sorting algorithms can be translated into algorithms for solving the more general problem of finding Hamilton paths in tournaments. [169, pp. 12-15]), which we now translate into the language of tournaments. For simplicity we shall assume that the number of vertices of the input tournament is a power of two. 70. It is convenient to state the algorithm as a recursive algorithm (which is the reason why we specify a parameter for the algorithm). We assume that the tournament is available through its adjacency matrix.

The minimum number, χ(H), of independent sets in a proper colouring of H is the chromatic number of H. 3, the operation of composition of digraphs was introduced. Considering complete biorientations of undirected graphs, one can easily define the operation of composition of undirected graphs. Let H be a graph with vertex set {v1 , v2 , . . , vn }, and let G1 , G2 , . . , Gn be graphs which are pairwise vertex-disjoint. The composition H[G1 , G2 , . . , Gn ] is the graph L with vertex set V (G1 ) ∪ V (G2 ) ∪ .

V (Dt ) = V (D) (recall that a digraph with only one vertex is strong). Moreover, we must have V (Di ) ∩ V (Dj ) = ∅ for every i = j as otherwise all the vertices V (Di ) ∪ V (Dj ) are reachable from each other, implying that the vertices of V (Di ) ∪ V (Dj ) belong to the same strong component of D. We call V (D1 ) ∪ . . ∪ V (Dt ) the strong decomposition of D. The strong component digraph SC(D) of D is obtained by contracting strong components of D and deleting any parallel arcs obtained in this process.

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