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Some remarks on the geodetic number of a graph
A set of vertices D of a graph G is geodetic if every vertex of G lies on a shortest path between two not necessarily distinct vertices in D. The geodetic number of G is the minimum cardinality of a geodetic set of G. We prove that it is NP complete to decide for a given chordal or chordal bipartite graph G and a given integer k whether G has a geodetic set of cardinality at most k. Furthermore, we prove an upper bound on the geodetic number of graphs without short cycles and study the geodetic number of cographs, split graphs, and unit interval graphs
A note on the convexity number for complementary prisms
In the geodetic convexity, a set of vertices of a graph is
if all vertices belonging to any shortest path between two
vertices of lie in . The cardinality of a maximum proper convex
set of is the of . The
of a graph arises from the
disjoint union of the graph and by adding the edges of a
perfect matching between the corresponding vertices of and .
In this work, we we prove that the decision problem related to the convexity
number is NP-complete even restricted to complementary prisms, we determine
when is disconnected or is a cograph, and we
present a lower bound when .Comment: 10 pages, 2 figure
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