3,203 research outputs found
Linear rank-width of distance-hereditary graphs II. Vertex-minor obstructions
In the companion paper [Linear rank-width of distance-hereditary graphs I. A
polynomial-time algorithm, Algorithmica 78(1):342--377, 2017], we presented a
characterization of the linear rank-width of distance-hereditary graphs, from
which we derived an algorithm to compute it in polynomial time. In this paper,
we investigate structural properties of distance-hereditary graphs based on
this characterization.
First, we prove that for a fixed tree , every distance-hereditary graph of
sufficiently large linear rank-width contains a vertex-minor isomorphic to .
We extend this property to bigger graph classes, namely, classes of graphs
whose prime induced subgraphs have bounded linear rank-width. Here, prime
graphs are graphs containing no splits. We conjecture that for every tree ,
every graph of sufficiently large linear rank-width contains a vertex-minor
isomorphic to . Our result implies that it is sufficient to prove this
conjecture for prime graphs.
For a class of graphs closed under taking vertex-minors, a graph
is called a vertex-minor obstruction for if but all of
its proper vertex-minors are contained in . Secondly, we provide, for
each , a set of distance-hereditary graphs that contains all
distance-hereditary vertex-minor obstructions for graphs of linear rank-width
at most . Also, we give a simpler way to obtain the known vertex-minor
obstructions for graphs of linear rank-width at most .Comment: 38 pages, 13 figures, 1 table, revised journal version. A preliminary
version of Section 5 appeared in the proceedings of WG1
On Brambles, Grid-Like Minors, and Parameterized Intractability of Monadic Second-Order Logic
Brambles were introduced as the dual notion to treewidth, one of the most
central concepts of the graph minor theory of Robertson and Seymour. Recently,
Grohe and Marx showed that there are graphs G, in which every bramble of order
larger than the square root of the treewidth is of exponential size in |G|. On
the positive side, they show the existence of polynomial-sized brambles of the
order of the square root of the treewidth, up to log factors. We provide the
first polynomial time algorithm to construct a bramble in general graphs and
achieve this bound, up to log-factors. We use this algorithm to construct
grid-like minors, a replacement structure for grid-minors recently introduced
by Reed and Wood, in polynomial time. Using the grid-like minors, we introduce
the notion of a perfect bramble and an algorithm to find one in polynomial
time. Perfect brambles are brambles with a particularly simple structure and
they also provide us with a subgraph that has bounded degree and still large
treewidth; we use them to obtain a meta-theorem on deciding certain
parameterized subgraph-closed problems on general graphs in time singly
exponential in the parameter.
The second part of our work deals with providing a lower bound to Courcelle's
famous theorem, stating that every graph property that can be expressed by a
sentence in monadic second-order logic (MSO), can be decided by a linear time
algorithm on classes of graphs of bounded treewidth. Using our results from the
first part of our work we establish a strong lower bound for tractability of
MSO on classes of colored graphs
Hitting minors, subdivisions, and immersions in tournaments
The Erd\H{o}s-P\'osa property relates parameters of covering and packing of
combinatorial structures and has been mostly studied in the setting of
undirected graphs. In this note, we use results of Chudnovsky, Fradkin, Kim,
and Seymour to show that, for every directed graph (resp.
strongly-connected directed graph ), the class of directed graphs that
contain as a strong minor (resp. butterfly minor, topological minor) has
the vertex-Erd\H{o}s-P\'osa property in the class of tournaments. We also prove
that if is a strongly-connected directed graph, the class of directed
graphs containing as an immersion has the edge-Erd\H{o}s-P\'osa property in
the class of tournaments.Comment: Accepted to Discrete Mathematics & Theoretical Computer Science.
Difference with the previous version: use of the DMTCS article class. For a
version with hyperlinks see the previous versio
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