5,813 research outputs found
Spanning trees short or small
We study the problem of finding small trees. Classical network design
problems are considered with the additional constraint that only a specified
number of nodes are required to be connected in the solution. A
prototypical example is the MST problem in which we require a tree of
minimum weight spanning at least nodes in an edge-weighted graph. We show
that the MST problem is NP-hard even for points in the Euclidean plane. We
provide approximation algorithms with performance ratio for the
general edge-weighted case and for the case of points in the
plane. Polynomial-time exact solutions are also presented for the class of
decomposable graphs which includes trees, series-parallel graphs, and bounded
bandwidth graphs, and for points on the boundary of a convex region in the
Euclidean plane. We also investigate the problem of finding short trees, and
more generally, that of finding networks with minimum diameter. A simple
technique is used to provide a polynomial-time solution for finding -trees
of minimum diameter. We identify easy and hard problems arising in finding
short networks using a framework due to T. C. Hu.Comment: 27 page
The Steiner diameter of a graph
The Steiner distance of a graph, introduced by Chartrand, Oellermann, Tian
and Zou in 1989, is a natural generalization of the concept of classical graph
distance. For a connected graph of order at least and , the \emph{Steiner distance} among the vertices of is the
minimum size among all connected subgraphs whose vertex sets contain . Let
be two integers with . Then the \emph{Steiner
-eccentricity } of a vertex of is defined by . Furthermore, the
\emph{Steiner -diameter} of is . In 2011, Chartrand, Okamoto and Zhang showed that . In this paper, graphs with are
characterized, respectively. We also consider the Nordhaus-Gaddum-type results
for the parameter . We determine sharp upper and lower bounds of
and
for a graph of order . Some graph classes attaining these bounds are
also given.Comment: 14 page
Total monochromatic connection of graphs
A graph is said to be {\it total-colored} if all the edges and the vertices
of the graph are colored. A path in a total-colored graph is a {\it total
monochromatic path} if all the edges and internal vertices on the path have the
same color. A total-coloring of a graph is a {\it total
monochromatically-connecting coloring} ({\it TMC-coloring}, for short) if any
two vertices of the graph are connected by a total monochromatic path of the
graph. For a connected graph , the {\it total monochromatic connection
number}, denoted by , is defined as the maximum number of colors used
in a TMC-coloring of . These concepts are inspired by the concepts of
monochromatic connection number , monochromatic vertex connection number
and total rainbow connection number of a connected graph .
Let denote the number of leaves of a tree , and let is a spanning tree of for a connected graph . In this
paper, we show that there are many graphs such that ,
and moreover, we prove that for almost all graphs ,
holds. Furthermore, we compare with and ,
respectively, and obtain that there exist graphs such that is not
less than and vice versa, and that holds for
almost all graphs. Finally, we prove that , and the
equality holds if and only if is a complete graph.Comment: 12 page
Characterizing -Distance Graphs and Solving the Equations or
Let be a finite, simple graph with vertex set . The -distance
graph of is the graph with the same vertex set as and two
vertices are adjacent if and only if their distance in is exactly . A
graph is a -distance graph if there exists a graph such that
. In this paper, we give three characterizations of -distance
graphs, and find all graphs such that or ,
where is an integer, is the path of order , and is the
complete graph of order
Proper connection numbers of complementary graphs
A path in an edge-colored graph is called a proper path if no two
adjacent edges of are colored the same, and is proper connected if
every two vertices of are connected by a proper path in . The proper
connection number of a connected graph , denoted by , is the minimum
number of colors that are needed to make proper connected. In this paper,
we investigate the proper connection number of the complement of graph
according to some constraints of itself. Also, we characterize the graphs
on vertices that have proper connection number . Using this result, we
give a Nordhaus-Gaddum-type theorem for the proper connection number. We prove
that if and are both connected, then , and the only graph attaining the upper bound is
the tree with maximum degree .Comment: 12 page
Nearly Tight Low Stretch Spanning Trees
We prove that any graph with points has a distribution
over spanning trees such that for any edge the expected stretch is bounded by . Our
result is obtained via a new approach of building ``highways'' between portals
and a new strong diameter probabilistic decomposition theorem
Tight Nordhaus-Gaddum-type upper bound for total-rainbow connection number of graphs
A graph is said to be \emph{total-colored} if all the edges and the vertices
of the graph are colored. A total-colored graph is \emph{total-rainbow
connected} if any two vertices of the graph are connected by a path whose edges
and internal vertices have distinct colors. For a connected graph , the
\emph{total-rainbow connection number} of , denoted by , is the
minimum number of colors required in a total-coloring of to make
total-rainbow connected. In this paper, we first characterize the graphs having
large total-rainbow connection numbers. Based on this, we obtain a
Nordhaus-Gaddum-type upper bound for the total-rainbow connection number. We
prove that if and are connected complementary graphs on
vertices, then when and
when . Examples are given to show that
the upper bounds are sharp for . This completely solves a conjecture
in [Y. Ma, Total rainbow connection number and complementary graph, Results in
Mathematics 70(1-2)(2016), 173-182].Comment: 20 page
Edge Coloring with Minimum Reload/Changeover Costs
In an edge-colored graph, a traversal cost occurs at a vertex along a path
when consecutive edges with different colors are traversed. The value of the
traversal cost depends only on the colors of the traversed edges. This concept
leads to two global cost measures, namely the \emph{reload cost} and the
\emph{changeover cost}, that have been studied in the literature and have
various applications in telecommunications, transportation networks, and energy
distribution networks. Previous work focused on problems with an edge-colored
graph being part of the input. In this paper, we formulate and focus on two
pairs of problems that aim to find an edge coloring of a graph so as to
minimize the reload and changeover costs. The first pair of problems aims to
find a proper edge coloring so that the reload/changeover cost of a set of
paths is minimized. The second pair of problems aim to find a proper edge
coloring and a spanning tree so that the reload/changeover cost is minimized.
We present several hardness results as well as polynomial-time solvable special
cases
Lower bounds for algebraic connectivity of graphs in terms of matching number or edge covering number
In this paper we characterize the unique graph whose algebraic connectivity
is minimum among all connected graphs with given order and fixed matching
number or edge covering number, and present two lower bounds for the algebraic
connectivity in terms of the matching number or edge covering number.Comment: arXiv admin note: substantial text overlap with arXiv:1310.853
Distance proper connection of graphs
Let be an edge-colored connected graph. A path in is called a
distance -proper path if no two edges of the same color appear with fewer
than edges in between on . The graph is called -proper
connected if every pair of distinct vertices of are connected by
pairwise internally vertex-disjoint distance -proper paths in . For a
-connected graph , the minimum number of colors needed to make
-proper connected is called the -proper connection number
of and denoted by . In this paper, we prove that
for any -connected graph . Considering graph
operations, we find that is a sharp upper bound for the -proper
connection number of the join and the Cartesian product of almost all graphs.
In addition, we find some basic properties of the -proper connection
number and determine the values of where is a traceable
graph, a tree, a complete bipartite graph, a complete multipartite graph, a
wheel, a cube or a permutation graph of a nontrivial traceable graph.Comment: 18 page
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