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Geodetic sets in graphs

Gary ChartrandFrank HararyPing Zhang — 2000

Discussiones Mathematicae Graph Theory

For two vertices u and v of a graph G, the closed interval I[u,v] consists of u, v, and all vertices lying in some u-v geodesic in G. If S is a set of vertices of G, then I[S] is the union of all sets I[u,v] for u, v ∈ S. If I[S] = V(G), then S is a geodetic set for G. The geodetic number g(G) is the minimum cardinality of a geodetic set. A set S of vertices in a graph G is uniform if the distance between every two distinct vertices of S is the same fixed number. A geodetic set is essential if for...

Destroying symmetry by orienting edges: complete graphs and complete bigraphs

Frank HararyMichael S. Jacobson — 2001

Discussiones Mathematicae Graph Theory

Our purpose is to introduce the concept of determining the smallest number of edges of a graph which can be oriented so that the resulting mixed graph has the trivial automorphism group. We find that this number for complete graphs is related to the number of identity oriented trees. For complete bipartite graphs K s , t , s ≤ t, this number does not always exist. We determine for s ≤ 4 the values of t for which this number does exist.

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