164,814 research outputs found
Unit Interval Editing is Fixed-Parameter Tractable
Given a graph~ and integers , , and~, the unit interval
editing problem asks whether can be transformed into a unit interval graph
by at most vertex deletions, edge deletions, and edge
additions. We give an algorithm solving this problem in time , where , and denote respectively
the numbers of vertices and edges of . Therefore, it is fixed-parameter
tractable parameterized by the total number of allowed operations.
Our algorithm implies the fixed-parameter tractability of the unit interval
edge deletion problem, for which we also present a more efficient algorithm
running in time . Another result is an -time algorithm for the unit interval vertex deletion problem,
significantly improving the algorithm of van 't Hof and Villanger, which runs
in time .Comment: An extended abstract of this paper has appeared in the proceedings of
ICALP 2015. Update: The proof of Lemma 4.2 has been completely rewritten; an
appendix is provided for a brief overview of related graph classe
Periods implying almost all periods, trees with snowflakes, and zero entropy maps
Let be a compact tree, be a continuous map from to itself,
be the number of endpoints and be the number of edges of .
We show that if has no prime divisors less than and has a
cycle of period , then has cycles of all periods greater than
and topological entropy ; so if is the least prime
number greater than and has cycles of all periods from 1 to
, then has cycles of all periods (this verifies a conjecture
of Misiurewicz for tree maps). Together with the spectral decomposition theorem
for graph maps it implies that iff there exists such that has
a cycle of period for any . We also define {\it snowflakes} for tree
maps and show that iff every cycle of is a snowflake or iff the
period of every cycle of is of form where is an odd
integer with prime divisors less than
Connectivity Oracles for Graphs Subject to Vertex Failures
We introduce new data structures for answering connectivity queries in graphs
subject to batched vertex failures. A deterministic structure processes a batch
of failed vertices in time and thereafter
answers connectivity queries in time. It occupies space . We develop a randomized Monte Carlo version of our data structure
with update time , query time , and space
for any failure bound . This is the first connectivity oracle for
general graphs that can efficiently deal with an unbounded number of vertex
failures.
We also develop a more efficient Monte Carlo edge-failure connectivity
oracle. Using space , edge failures are processed in time and thereafter, connectivity queries are answered in
time, which are correct w.h.p.
Our data structures are based on a new decomposition theorem for an
undirected graph , which is of independent interest. It states that
for any terminal set we can remove a set of
vertices such that the remaining graph contains a Steiner forest for with
maximum degree
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