4,407 research outputs found
Avoidability beyond paths
The concept of avoidable paths in graphs was introduced by Beisegel,
Chudnovsky, Gurvich, Milani\v{c}, and Servatius in 2019 as a common
generalization of avoidable vertices and simplicial paths. In 2020, Bonamy,
Defrain, Hatzel, and Thiebaut proved that every graph containing an induced
path of order also contains an avoidable induced path of the same order.
They also asked whether one could generalize this result to other avoidable
structures, leaving the notion of avoidability up to interpretation. In this
paper we address this question: we specify the concept of avoidability for
arbitrary graphs equipped with two terminal vertices. We provide both positive
and negative results, some of which appear to be related to the recent work by
Chudnovsky, Norin, Seymour, and Turcotte [arXiv:2301.13175]
Metro-Line Crossing Minimization: Hardness, Approximations, and Tractable Cases
Crossing minimization is one of the central problems in graph drawing.
Recently, there has been an increased interest in the problem of minimizing
crossings between paths in drawings of graphs. This is the metro-line crossing
minimization problem (MLCM): Given an embedded graph and a set L of simple
paths, called lines, order the lines on each edge so that the total number of
crossings is minimized. So far, the complexity of MLCM has been an open
problem. In contrast, the problem variant in which line ends must be placed in
outermost position on their edges (MLCM-P) is known to be NP-hard. Our main
results answer two open questions: (i) We show that MLCM is NP-hard. (ii) We
give an -approximation algorithm for MLCM-P
Optimal redundancy against disjoint vulnerabilities in networks
Redundancy is commonly used to guarantee continued functionality in networked
systems. However, often many nodes are vulnerable to the same failure or
adversary. A "backup" path is not sufficient if both paths depend on nodes
which share a vulnerability.For example, if two nodes of the Internet cannot be
connected without using routers belonging to a given untrusted entity, then all
of their communication-regardless of the specific paths utilized-will be
intercepted by the controlling entity.In this and many other cases, the
vulnerabilities affecting the network are disjoint: each node has exactly one
vulnerability but the same vulnerability can affect many nodes. To discover
optimal redundancy in this scenario, we describe each vulnerability as a color
and develop a "color-avoiding percolation" which uncovers a hidden
color-avoiding connectivity. We present algorithms for color-avoiding
percolation of general networks and an analytic theory for random graphs with
uniformly distributed colors including critical phenomena. We demonstrate our
theory by uncovering the hidden color-avoiding connectivity of the Internet. We
find that less well-connected countries are more likely able to communicate
securely through optimally redundant paths than highly connected countries like
the US. Our results reveal a new layer of hidden structure in complex systems
and can enhance security and robustness through optimal redundancy in a wide
range of systems including biological, economic and communications networks.Comment: 15 page
Conformal invariance of crossing probabilities for the Ising model with free boundary conditions
We prove that crossing probabilities for the critical planar Ising model with
free boundary conditions are conformally invariant in the scaling limit, a
phenomenon first investigated numerically by Langlands, Lewis and Saint-Aubin.
We do so by establishing the convergence of certain exploration processes
towards SLE. We also construct an exploration tree
for free boundary conditions, analogous to the one introduced by Sheffield.Comment: 18 pages, 4 figures, v2: journal versio
Ordering Metro Lines by Block Crossings
A problem that arises in drawings of transportation networks is to minimize
the number of crossings between different transportation lines. While this can
be done efficiently under specific constraints, not all solutions are visually
equivalent. We suggest merging crossings into block crossings, that is,
crossings of two neighboring groups of consecutive lines. Unfortunately,
minimizing the total number of block crossings is NP-hard even for very simple
graphs. We give approximation algorithms for special classes of graphs and an
asymptotically worst-case optimal algorithm for block crossings on general
graphs. That is, we bound the number of block crossings that our algorithm
needs and construct worst-case instances on which the number of block crossings
that is necessary in any solution is asymptotically the same as our bound
Light subgraphs in graphs with average degree at most four
A graph is said to be {\em light} in a family of graphs if
at least one member of contains a copy of and there exists
an integer such that each member of
with a copy of also has a copy of such that
for all . In this
paper, we study the light graphs in the class of graphs with small average
degree, including the plane graphs with some restrictions on girth.Comment: 12 pages, 18 figure
Online version of the theorem of Thue
A sequence S is nonrepetitive if no two adjacent blocks of S are the same. In
1906 Thue proved that there exist arbitrarily long nonrepetitive sequences over
3 symbols. We consider the online variant of this result in which a
nonrepetitive sequence is constructed during a play between two players: Bob is
choosing a position in a sequence and Alice is inserting a symbol on that
position taken from a fixed set A. The goal of Bob is to force Alice to create
a repetition, and if he succeeds, then the game stops. The goal of Alice is
naturally to avoid that and thereby to construct a nonrepetitive sequence of
any given length. We prove that Alice has a strategy to play arbitrarily long
provided the size of the set A is at least 12. This is the online version of
the Theorem of Thue. The proof is based on nonrepetitive colorings of
outerplanar graphs. On the other hand, one can prove that even over 4 symbols
Alice has no chance to play for too long. The minimum size of the set of
symbols needed for the online version of Thue's theorem remains unknown
Planar Ramsey graphs
We say that a graph is planar unavoidable if there is a planar graph
such that any red/blue coloring of the edges of contains a monochromatic
copy of , otherwise we say that is planar avoidable. I.e., is planar
unavoidable if there is a Ramsey graph for that is planar. It follows from
the Four-Color Theorem and a result of Gon\c{c}alves that if a graph is planar
unavoidable then it is bipartite and outerplanar. We prove that the cycle on
vertices and any path are planar unavoidable. In addition, we prove that
all trees of radius at most are planar unavoidable and there are trees of
radius that are planar avoidable. We also address the planar unavoidable
notion in more than two colors
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