28 research outputs found
Finding topological subgraphs is fixed-parameter tractable
We show that for every fixed undirected graph , there is a
time algorithm that tests, given a graph , if contains as a
topological subgraph (that is, a subdivision of is subgraph of ). This
shows that topological subgraph testing is fixed-parameter tractable, resolving
a longstanding open question of Downey and Fellows from 1992. As a corollary,
for every we obtain an time algorithm that tests if there is
an immersion of into a given graph . This answers another open question
raised by Downey and Fellows in 1992
On retracts, absolute retracts, and folds in cographs
Let G and H be two cographs. We show that the problem to determine whether H
is a retract of G is NP-complete. We show that this problem is fixed-parameter
tractable when parameterized by the size of H. When restricted to the class of
threshold graphs or to the class of trivially perfect graphs, the problem
becomes tractable in polynomial time. The problem is also soluble when one
cograph is given as an induced subgraph of the other. We characterize absolute
retracts of cographs.Comment: 15 page
Parameterized Algorithms for Min-Max Multiway Cut and List Digraph Homomorphism
In this paper we design {\sf FPT}-algorithms for two parameterized problems.
The first is \textsc{List Digraph Homomorphism}: given two digraphs and
and a list of allowed vertices of for every vertex of , the question is
whether there exists a homomorphism from to respecting the list
constraints. The second problem is a variant of \textsc{Multiway Cut}, namely
\textsc{Min-Max Multiway Cut}: given a graph , a non-negative integer
, and a set of terminals, the question is whether we can
partition the vertices of into parts such that (a) each part contains
one terminal and (b) there are at most edges with only one endpoint in
this part. We parameterize \textsc{List Digraph Homomorphism} by the number
of edges of that are mapped to non-loop edges of and we give a time
algorithm, where is the order of the host graph . We also prove that
\textsc{Min-Max Multiway Cut} can be solved in time . Our approach introduces a general problem, called
{\sc List Allocation}, whose expressive power permits the design of
parameterized reductions of both aforementioned problems to it. Then our
results are based on an {\sf FPT}-algorithm for the {\sc List Allocation}
problem that is designed using a suitable adaptation of the {\em randomized
contractions} technique (introduced by [Chitnis, Cygan, Hajiaghayi, Pilipczuk,
and Pilipczuk, FOCS 2012]).Comment: An extended abstract of this work will appear in the Proceedings of
the 10th International Symposium on Parameterized and Exact Computation
(IPEC), Patras, Greece, September 201
Everything you always wanted to know about the parameterized complexity of Subgraph Isomorphism (but were afraid to ask)
Given two graphs H and G, the Subgraph Isomorphism problem asks if H is isomorphic to a subgraph of G. While NP-hard in general, algorithms exist for various parameterized versions of the problem. However, the literature contains very little guidance on which combinations of parameters can or cannot be exploited algorithmically. Our goal is to systematically investigate the possible parameterized algorithms that can exist for Subgraph Isomorphism.
We develop a framework involving 10 relevant parameters for each of H and G (such as treewidth, pathwidth, genus, maximum degree, number of vertices, number of components, etc.), and ask if an algorithm with running time f1_(p_1,p_2,...,p_l).n^f_2(p_(l+1),...,p_k) exists, where each of p_1,...,p_k is one of the 10 parameters depending only on H or G. We show that all the questions arising in this framework are answered by a set of 11 maximal positive results (algorithms) and a set of 17 maximal negative results (hardness proofs); some of these results already appear in the literature, while others are new in this paper.
On the algorithmic side, our study reveals for example that an unexpected combination of bounded degree, genus, and feedback vertex set number of G gives rise to a highly nontrivial algorithm for Subgraph Isomorphism. On the hardness side, we present W[1]-hardness proofs under extremely restricted conditions, such as when H is a bounded-degree tree of constant pathwidth and G is a planar graph of bounded pathwidth
A more accurate view of the Flat Wall Theorem
We introduce a supporting combinatorial framework for the Flat Wall Theorem.
In particular, we suggest two variants of the theorem and we introduce a new,
more versatile, concept of wall homogeneity as well as the notion of regularity
in flat walls. All proposed concepts and results aim at facilitating the use of
the irrelevant vertex technique in future algorithmic applications.Comment: arXiv admin note: text overlap with arXiv:2004.1269
Forbidding Kuratowski Graphs as Immersions
The immersion relation is a partial ordering relation on graphs that is
weaker than the topological minor relation in the sense that if a graph
contains a graph as a topological minor, then it also contains it as an
immersion but not vice versa. Kuratowski graphs, namely and ,
give a precise characterization of planar graphs when excluded as topological
minors. In this note we give a structural characterization of the graphs that
exclude Kuratowski graphs as immersions. We prove that they can be constructed
by applying consecutive -edge-sums, for , starting from graphs that
are planar sub-cubic or of branch-width at most 10