16,316 research outputs found
Minimum Weight Resolving Sets of Grid Graphs
For a simple graph and for a pair of vertices , we say
that a vertex resolves and if the shortest path from to
is of a different length than the shortest path from to . A set of
vertices is a resolving set if for every pair of vertices
and in , there exists a vertex that resolves and . The
minimum weight resolving set problem is to find a resolving set for a
weighted graph such that is minimum, where is
the weight of vertex . In this paper, we explore the possible solutions of
this problem for grid graphs where . We give
a complete characterisation of solutions whose cardinalities are 2 or 3, and
show that the maximum cardinality of a solution is . We also provide a
characterisation of a class of minimals whose cardinalities range from to
.Comment: 21 pages, 10 figure
On the Metric Dimension of Cartesian Products of Graphs
A set S of vertices in a graph G resolves G if every vertex is uniquely
determined by its vector of distances to the vertices in S. The metric
dimension of G is the minimum cardinality of a resolving set of G. This paper
studies the metric dimension of cartesian products G*H. We prove that the
metric dimension of G*G is tied in a strong sense to the minimum order of a
so-called doubly resolving set in G. Using bounds on the order of doubly
resolving sets, we establish bounds on G*H for many examples of G and H. One of
our main results is a family of graphs G with bounded metric dimension for
which the metric dimension of G*G is unbounded
Stabbing line segments with disks: complexity and approximation algorithms
Computational complexity and approximation algorithms are reported for a
problem of stabbing a set of straight line segments with the least cardinality
set of disks of fixed radii where the set of segments forms a straight
line drawing of a planar graph without edge crossings. Close
geometric problems arise in network security applications. We give strong
NP-hardness of the problem for edge sets of Delaunay triangulations, Gabriel
graphs and other subgraphs (which are often used in network design) for and some constant where and
are Euclidean lengths of the longest and shortest graph edges
respectively. Fast -time -approximation algorithm is
proposed within the class of straight line drawings of planar graphs for which
the inequality holds uniformly for some constant
i.e. when lengths of edges of are uniformly bounded from above by
some linear function of Comment: 12 pages, 1 appendix, 15 bibliography items, 6th International
Conference on Analysis of Images, Social Networks and Texts (AIST-2017
Polynomial sequences of binomial-type arising in graph theory
In this paper, we show that the solution to a large class of "tiling"
problems is given by a polynomial sequence of binomial type. More specifically,
we show that the number of ways to place a fixed set of polyominos on an
toroidal chessboard such that no two polyominos overlap is
eventually a polynomial in , and that certain sets of these polynomials
satisfy binomial-type recurrences. We exhibit generalizations of this theorem
to higher dimensions and other lattices. Finally, we apply the techniques
developed in this paper to resolve an open question about the structure of
coefficients of chromatic polynomials of certain grid graphs (namely that they
also satisfy a binomial-type recurrence).Comment: 15 page
On Routing Disjoint Paths in Bounded Treewidth Graphs
We study the problem of routing on disjoint paths in bounded treewidth graphs
with both edge and node capacities. The input consists of a capacitated graph
and a collection of source-destination pairs . The goal is to maximize the number of pairs that
can be routed subject to the capacities in the graph. A routing of a subset
of the pairs is a collection of paths such that,
for each pair , there is a path in
connecting to . In the Maximum Edge Disjoint Paths (MaxEDP) problem,
the graph has capacities on the edges and a routing
is feasible if each edge is in at most of
the paths of . The Maximum Node Disjoint Paths (MaxNDP) problem is
the node-capacitated counterpart of MaxEDP.
In this paper we obtain an approximation for MaxEDP on graphs of
treewidth at most and a matching approximation for MaxNDP on graphs of
pathwidth at most . Our results build on and significantly improve the work
by Chekuri et al. [ICALP 2013] who obtained an approximation
for MaxEDP
Colored Non-Crossing Euclidean Steiner Forest
Given a set of -colored points in the plane, we consider the problem of
finding trees such that each tree connects all points of one color class,
no two trees cross, and the total edge length of the trees is minimized. For
, this is the well-known Euclidean Steiner tree problem. For general ,
a -approximation algorithm is known, where is the
Steiner ratio.
We present a PTAS for , a -approximation algorithm
for , and two approximation algorithms for general~, with ratios
and
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