2,400 research outputs found
Complexity of Token Swapping and its Variants
In the Token Swapping problem we are given a graph with a token placed on
each vertex. Each token has exactly one destination vertex, and we try to move
all the tokens to their destinations, using the minimum number of swaps, i.e.,
operations of exchanging the tokens on two adjacent vertices. As the main
result of this paper, we show that Token Swapping is -hard parameterized
by the length of a shortest sequence of swaps. In fact, we prove that, for
any computable function , it cannot be solved in time where is the number of vertices of the input graph, unless the ETH
fails. This lower bound almost matches the trivial -time algorithm.
We also consider two generalizations of the Token Swapping, namely Colored
Token Swapping (where the tokens have different colors and tokens of the same
color are indistinguishable), and Subset Token Swapping (where each token has a
set of possible destinations). To complement the hardness result, we prove that
even the most general variant, Subset Token Swapping, is FPT in nowhere-dense
graph classes.
Finally, we consider the complexities of all three problems in very
restricted classes of graphs: graphs of bounded treewidth and diameter, stars,
cliques, and paths, trying to identify the borderlines between polynomial and
NP-hard cases.Comment: 23 pages, 7 Figure
On groups generated by two positive multi-twists: Teichmueller curves and Lehmer's number
From a simple observation about a construction of Thurston, we derive several
interesting facts about subgroups of the mapping class group generated by two
positive multi-twists. In particular, we identify all configurations of curves
for which the corresponding groups fail to be free, and show that a subset of
these determine the same set of Teichmueller curves as the non-obtuse lattice
triangles which were classified by Kenyon, Smillie, and Puchta. We also
identify a pseudo-Anosov automorphism whose dilatation is Lehmer's number, and
show that this is minimal for the groups under consideration. In addition, we
describe a connection to work of McMullen on Coxeter groups and related work of
Hironaka on a construction of an interesting class of fibered links.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol8/paper36.abs.htm
Subgraph statistics in subcritical graph classes
Let H be a fixed graph and math formula a subcritical graph class. In this paper we show that the number of occurrences of H (as a subgraph) in a graph in math formula of order n, chosen uniformly at random, follows a normal limiting distribution with linear expectation and variance. The main ingredient in our proof is the analytic framework developed by Drmota, Gittenberger and Morgenbesser to deal with infinite systems of functional equations [Drmota, Gittenberger, and Morgenbesser, Submitted]. As a case study, we obtain explicit expressions for the number of triangles and cycles of length 4 in the family of series-parallel graphs.Postprint (author's final draft
P_4-Decomposability in Regular Graphs and Multigraphs
The main objective of this thesis is to review and expand the study of graph decomposability. An H-decomposition of a graph G=(V,E) is a partitioning of the edge set, , into edge-disjoint isomorphic copies of a subgraph H. In particular we focus on the decompositions of graphs into paths. We prove that a 2,4 mutligraph with maximum multiplicity 2 admits a C_2,C_3-free Euler tour (and thus, a decomposition into paths of length 3 if it has size a multiple of 3) if and only if it avoids a set of 15 forbidden structures. We also prove that a 4-regular multigraph with maximum multiplicity 2 admits a decomposition into paths of length three if and only if it has size a multiple of 3 and no three vertices induce more than 4 edges. We go on to outline drafted work reflecting further research into path decomposition problems
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