The upper forcing edge-to-vertex geodetic number of a graph
نویسنده
چکیده
For a connected graph G = (V,E), a set S ⊆ E is called an edge-to-vertex geodetic set of G if every vertex of G is either incident with an edge of S or lies on a geodesic joining some pair of edges of S. The minimum cardinality of an edge-to-vertex geodetic set of G is gev(G). Any edge-to-vertex geodetic set of cardinality gev(G) is called an edge-to-vertex geodetic basis of G. A subset T ⊆ S is called a forcing subset for S if S is the unique minimum edge-to-vertex geodetic set containing T . A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing edge-to-vertex geodetic number of S, denoted by fev(S), is the cardinality of a minimum forcing subset of S. The upper forcing edge-to-vertex geodetic number of G, denoted by f ev(G), is f ev(G) = max {fev(S)}, where the maximum is taken over all minimum edge-to-vertex geodetic sets S in G. It is shown that the upper forcing edge-to-vertex geodetic number lies between 0 and gev(G). Also, the upper forcing edge-to-vertex geodetic number of certain classes of graphs such as cycle, tree, complete graph and complete bipartite graph are determined.
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