2,034 research outputs found
On graph equivalences preserved under extensions
Let R be an equivalence relation on graphs. By the strengthening of R we mean
the relation R' such that graphs G and H are in the relation R' if for every
graph F, the union of the graphs G and F is in the relation R with the union of
the graphs H and F. We study strengthenings of equivalence relations on graphs.
The most important case that we consider concerns equivalence relations defined
by graph properties. We obtain results on the strengthening of equivalence
relations determined by the properties such as being a k-connected graph,
k-colorable, hamiltonian and planar
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
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