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Strong geodetic problem on Cartesian products of graphs
The strong geodetic problem is a recent variation of the geodetic problem.
For a graph , its strong geodetic number is the cardinality of
a smallest vertex subset , such that each vertex of lies on a fixed
shortest path between a pair of vertices from . In this paper, the strong
geodetic problem is studied on the Cartesian product of graphs. A general upper
bound for is determined, as well as exact values
for , , and certain prisms.
Connections between the strong geodetic number of a graph and its subgraphs are
also discussed.Comment: 18 pages, 9 figure
On the Computational Complexity of the Strong Geodetic Recognition Problem
A strong geodetic set of a graph~ is a vertex set~
in which it is possible to cover all the remaining vertices of~ by assigning a unique shortest path between each vertex pair of~. In the
Strong Geodetic problem (SG) a graph~ and a positive integer~ are given
as input and one has to decide whether~ has a strong geodetic set of
cardinality at most~. This problem is known to be NP-hard for general
graphs. In this work we introduce the Strong Geodetic Recognition problem
(SGR), which consists in determining whether even a given vertex set~ is strong geodetic. We demonstrate that this version is
NP-complete. We investigate and compare the computational complexity of both
decision problems restricted to some graph classes, deriving polynomial-time
algorithms, NP-completeness proofs, and initial parameterized complexity
results, including an answer to an open question in the literature for the
complexity of SG for chordal graphs
Computing Minimum Rainbow and Strong Rainbow Colorings of Block Graphs
A path in an edge-colored graph is rainbow if no two edges of it are
colored the same. The graph is rainbow-connected if there is a rainbow path
between every pair of vertices. If there is a rainbow shortest path between
every pair of vertices, the graph is strongly rainbow-connected. The
minimum number of colors needed to make rainbow-connected is known as the
rainbow connection number of , and is denoted by . Similarly,
the minimum number of colors needed to make strongly rainbow-connected is
known as the strong rainbow connection number of , and is denoted by
. We prove that for every , deciding whether
is NP-complete for split graphs, which form a subclass
of chordal graphs. Furthermore, there exists no polynomial-time algorithm for
approximating the strong rainbow connection number of an -vertex split graph
with a factor of for any unless P = NP. We
then turn our attention to block graphs, which also form a subclass of chordal
graphs. We determine the strong rainbow connection number of block graphs, and
show it can be computed in linear time. Finally, we provide a polynomial-time
characterization of bridgeless block graphs with rainbow connection number at
most 4.Comment: 13 pages, 3 figure
Hardness and approximation for the geodetic set problem in some graph classes
In this paper, we study the computational complexity of finding the
\emph{geodetic number} of graphs. A set of vertices of a graph is a
\emph{geodetic set} if any vertex of lies in some shortest path between
some pair of vertices from . The \textsc{Minimum Geodetic Set (MGS)} problem
is to find a geodetic set with minimum cardinality. In this paper, we prove
that solving the \textsc{MGS} problem is NP-hard on planar graphs with a
maximum degree six and line graphs. We also show that unless , there is
no polynomial time algorithm to solve the \textsc{MGS} problem with
sublogarithmic approximation factor (in terms of the number of vertices) even
on graphs with diameter . On the positive side, we give an
-approximation algorithm for the \textsc{MGS}
problem on general graphs of order . We also give a -approximation
algorithm for the \textsc{MGS} problem on the family of solid grid graphs which
is a subclass of planar graphs
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