8 research outputs found
New Hardness Results in Rainbow Connectivity
A path in an edge colored graph is said to be a rainbow path if no two edges
on the path have the same color. An edge colored graph is (strongly) rainbow
connected if there exists a (geodesic) rainbow path between every pair of
vertices. The (strong) rainbow connectivity of a graph , denoted by
(, respectively) is the smallest number of colors required to
edge color the graph such that the graph is (strong) rainbow connected. It is
known that for \emph{even} to decide whether the rainbow connectivity of a
graph is at most or not is NP-hard. It was conjectured that for all , to
decide whether is NP-hard. In this paper we prove this
conjecture. We also show that it is NP-hard to decide whether
or not even when is a bipartite graph.Comment: 15 pages, 2 figure
Strong rainbow connection numbers of toroidal meshes
In 2011, Li et al. \cite{LLL} obtained an upper bound of the strong rainbow
connection number of an -dimensional undirected toroidal mesh. In this
paper, this bound is improved. As a result, we give a negative answer to their
problem.Comment: 9 pages, 3 figure
Hardness result for the total rainbow -connection of graphs
A path in a total-colored graph is called \emph{total rainbow} if its edges
and internal vertices have distinct colors. For an -connected graph
and an integer with , the \emph{total rainbow
-connection number} of , denoted by , is the minimum number of
colors used in a total coloring of to make \emph{total rainbow
-connected}, that is, any two vertices of are connected by
internally vertex-disjoint total rainbow paths. In this paper, we study the
computational complexity of total rainbow -connection number of graphs. We
show that it is NP-complete to decide whether .Comment: 10 page
Rainbow connections for planar graphs and line graphs
An edge-colored graph is rainbow connected if any two vertices are
connected by a path whose edges have distinct colors. The rainbow connection
number of a connected graph , denoted by , is the smallest number of
colors that are needed in order to make rainbow connected. It was proved
that computing is an NP-Hard problem, as well as that even deciding
whether a graph has is NP-Complete. It is known that deciding whether
a given edge-colored graph is rainbow connected is NP-Complete. We will prove
that it is still NP-Complete even when the edge-colored graph is a planar
bipartite graph. We also give upper bounds of the rainbow connection number of
outerplanar graphs with small diameters. A vertex-colored graph is rainbow
vertex-connected if any two vertices are connected by a path whose internal
vertices have distinct colors. The rainbow vertex-connection number of a
connected graph , denoted by , is the smallest number of colors that
are needed in order to make rainbow vertex-connected. It is known that
deciding whether a given vertex-colored graph is rainbow vertex-connected is
NP-Complete. We will prove that it is still NP-Complete even when the
vertex-colored graph is a line graph.Comment: 13 page
Conflict-free connections: algorithm and complexity
A path in an(a) edge(vertex)-colored graph is called \emph{a conflict-free
path} if there exists a color used on only one of its edges(vertices). An(A)
edge(vertex)-colored graph is called \emph{conflict-free (vertex-)connected} if
there is a conflict-free path between each pair of distinct vertices. We call
the graph \emph{strongly conflict-free connected }if there exists a
conflict-free path of length for every two vertices .
And the \emph{strong conflict-free connection number} of a connected graph ,
denoted by , is defined as the smallest number of colors that are
required to make strongly conflict-free connected. In this paper, we first
investigate the question: Given a connected graph and a coloring $c: E(or\
V)\rightarrow \{1,2,\cdots,k\} \ (k\geq 1)Gc(k\geq 2)G(u,v)\in P \ (P\subset V\times V)scfc(G)\leq k(k\geq 2)Gk=2$, which was
not done in the old versio
Note on the complexity of deciding the rainbow connectedness for bipartite graphs
A path in an edge-colored graph is said to be a rainbow path if no two edges
on the path have the same color. An edge-colored graph is (strongly) rainbow
connected if there exists a rainbow (geodesic) path between every pair of
vertices. The (strong) rainbow connection number of , denoted by (,
respectively) , is the smallest number of colors that are needed in
order to make (strongly) rainbow connected. Though for a general graph
it is NP-Complete to decide whether , in this paper, we show that the
problem becomes easy when is a bipartite graph. Moreover, it is known that
deciding whether a given edge-colored (with an unbound number of colors) graph
is rainbow connected is NP-Complete. We will prove that it is still NP-Complete
even when the edge-colored graph is bipartite. We also show that a few NP-hard
problems on rainbow connection are indeed NP-Complete.Comment: 6 page
A Polynomial Time Algorithm for Computing the Strong Rainbow Connection Numbers of Odd Cacti
We consider the problem of computing the strong rainbow connection number
for cactus graphs in which all cycles have odd length. We present
a formula to calculate for such odd cacti which can be evaluated in
linear time, as well as an algorithm for computing the corresponding optimal
strong rainbow edge coloring, with polynomial worst case run time complexity.
Although computing is NP-hard in general, previous work has
demonstrated that it may be computed in polynomial time for certain classes of
graphs, including cycles, trees and block clique graphs. This work extends the
class of graphs for which may be computed in polynomial time.Comment: 18 pages, 4 figure
An integer program and new lower bounds for computing the strong rainbow connection numbers of graphs
We present an integer programming model to compute the strong rainbow
connection number, , of any simple graph . We introduce several
enhancements to the proposed model, including a fast heuristic, and a variable
elimination scheme. Moreover, we present a novel lower bound for which
may be of independent research interest. We solve the integer program both
directly and using an alternative method based on iterative lower bound
improvement, the latter of which we show to be highly effective in practice. To
our knowledge, these are the first computational methods for the strong rainbow
connection problem. We demonstrate the efficacy of our methods by computing the
strong rainbow connection numbers of graphs containing up to vertices.Comment: 27 pages, 7 figure