Mutual-visibility problems on graphs of diameter two
CoRR(2024)
摘要
The mutual-visibility problem in a graph G asks for the cardinality of a
largest set of vertices S⊆ V(G) so that for any two vertices x,y∈
S there is a shortest x,y-path P so that all internal vertices of P are
not in S. This is also said as x,y are visible with respect to S, or
S-visible for short. Variations of this problem are known, based on the
extension of the visibility property of vertices that are in and/or outside
S. Such variations are called total, outer and dual mutual-visibility
problems. This work is focused on studying the corresponding four visibility
parameters in graphs of diameter two, throughout showing bounds and/or closed
formulae for these parameters.
The mutual-visibility problem in the Cartesian product of two complete graphs
is equivalent to (an instance of) the celebrated Zarankievicz's problem. Here
we study the dual and outer mutual-visibility problem for the Cartesian product
of two complete graphs and all the mutual-visibility problems for the direct
product of such graphs as well. We also study all the mutual-visibility
problems for the line graphs of complete and complete bipartite graphs. As a
consequence of this study, we present several relationships between the
mentioned problems and some instances of the classical Turán problem.
Moreover, we study the visibility problems for cographs and several non-trivial
diameter-two graphs of minimum size.
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