16 research outputs found
Rank-width and Tree-width of H-minor-free Graphs
We prove that for any fixed r>=2, the tree-width of graphs not containing K_r
as a topological minor (resp. as a subgraph) is bounded by a linear (resp.
polynomial) function of their rank-width. We also present refinements of our
bounds for other graph classes such as K_r-minor free graphs and graphs of
bounded genus.Comment: 17 page
Reconfiguration on sparse graphs
A vertex-subset graph problem Q defines which subsets of the vertices of an
input graph are feasible solutions. A reconfiguration variant of a
vertex-subset problem asks, given two feasible solutions S and T of size k,
whether it is possible to transform S into T by a sequence of vertex additions
and deletions such that each intermediate set is also a feasible solution of
size bounded by k. We study reconfiguration variants of two classical
vertex-subset problems, namely Independent Set and Dominating Set. We denote
the former by ISR and the latter by DSR. Both ISR and DSR are PSPACE-complete
on graphs of bounded bandwidth and W[1]-hard parameterized by k on general
graphs. We show that ISR is fixed-parameter tractable parameterized by k when
the input graph is of bounded degeneracy or nowhere-dense. As a corollary, we
answer positively an open question concerning the parameterized complexity of
the problem on graphs of bounded treewidth. Moreover, our techniques generalize
recent results showing that ISR is fixed-parameter tractable on planar graphs
and graphs of bounded degree. For DSR, we show the problem fixed-parameter
tractable parameterized by k when the input graph does not contain large
bicliques, a class of graphs which includes graphs of bounded degeneracy and
nowhere-dense graphs
Graph sharing games: complexity and connectivity
We study the following combinatorial game played by two players, Alice and
Bob, which generalizes the Pizza game considered by Brown, Winkler and others.
Given a connected graph G with nonnegative weights assigned to its vertices,
the players alternately take one vertex of G in each turn. The first turn is
Alice's. The vertices are to be taken according to one (or both) of the
following two rules: (T) the subgraph of G induced by the taken vertices is
connected during the whole game, (R) the subgraph of G induced by the remaining
vertices is connected during the whole game. We show that if rules (T) and/or
(R) are required then for every epsilon > 0 and for every positive integer k
there is a k-connected graph G for which Bob has a strategy to obtain
(1-epsilon) of the total weight of the vertices. This contrasts with the
original Pizza game played on a cycle, where Alice is known to have a strategy
to obtain 4/9 of the total weight.
We show that the problem of deciding whether Alice has a winning strategy
(i.e., a strategy to obtain more than half of the total weight) is
PSPACE-complete if condition (R) or both conditions (T) and (R) are required.
We also consider a game played on connected graphs (without weights) where the
first player who violates condition (T) or (R) loses the game. We show that
deciding who has the winning strategy is PSPACE-complete.Comment: 22 pages, 11 figures; updated references, minor stylistical change
Tight query complexity bounds for learning graph partitions
Given a partition of a graph into connected components, the membership oracle
asserts whether any two vertices of the graph lie in the same component or not.
We prove that for , learning the components of an -vertex
hidden graph with components requires at least
membership queries. Our result improves on the best known information-theoretic
bound of queries, and exactly matches the query complexity of
the algorithm introduced by [Reyzin and Srivastava, 2007] for this problem.
Additionally, we introduce an oracle, with access to which one can learn the
number of components of in asymptotically fewer queries than learning the
full partition, thus answering another question posed by the same authors.
Lastly, we introduce a more applicable version of this oracle, and prove
asymptotically tight bounds of queries for both learning
and verifying an -edge hidden graph using it.Comment: Accepted for presentation at the 35th Annual Conference of Learning
Theory, 202
Testing first-order properties for subclasses of sparse graphs
We present a linear-time algorithm for deciding first-order (FO) properties
in classes of graphs with bounded expansion, a notion recently introduced by
Nesetril and Ossona de Mendez. This generalizes several results from the
literature, because many natural classes of graphs have bounded expansion:
graphs of bounded tree-width, all proper minor-closed classes of graphs, graphs
of bounded degree, graphs with no subgraph isomorphic to a subdivision of a
fixed graph, and graphs that can be drawn in a fixed surface in such a way that
each edge crosses at most a constant number of other edges. We deduce that
there is an almost linear-time algorithm for deciding FO properties in classes
of graphs with locally bounded expansion.
More generally, we design a dynamic data structure for graphs belonging to a
fixed class of graphs of bounded expansion. After a linear-time initialization
the data structure allows us to test an FO property in constant time, and the
data structure can be updated in constant time after addition/deletion of an
edge, provided the list of possible edges to be added is known in advance and
their simultaneous addition results in a graph in the class. All our results
also hold for relational structures and are based on the seminal result of
Nesetril and Ossona de Mendez on the existence of low tree-depth colorings
Testing first-order properties for subclasses of sparse graphs
We present a linear-time algorithm for deciding first-order (FO) properties in classes of graphs with bounded expansion, a notion recently introduced by NeÅ”etÅil and Ossona de Mendez. This generalizes several results from the literature, because many natural classes of graphs have bounded expansion: graphs of bounded tree-width, all proper minor-closed classes of graphs, graphs of bounded degree, graphs with no subgraph isomorphic to a subdivision of a fixed graph, and graphs that can be drawn in a fixed surface in such a way that each edge crosses at most a constant number of other edges. We deduce that there is an almost linear-time algorithm for deciding FO properties in classes of graphs with locally bounded expansion.
More generally, we design a dynamic data structure for graphs belonging to a fixed class of graphs of bounded expansion. After a linear-time initialization the data structure allows us to test an FO property in constant time, and the data structure can be updated in constant time after addition/deletion of an edge, provided the list of possible edges to be added is known in advance and their simultaneous addition results in a graph in the class. All our results also hold for relational structures and are based on the seminal result of NeÅ”etÅil and Ossona de Mendez on the existence of low tree-depth colorings
Domination Problems in Nowhere-Dense Classes
We investigate the parameterized complexity of generalisations and variations
of the dominating set problem on classes of graphs that are nowhere dense. In
particular, we show that the distance- dominating-set problem, also known
as the -centres problem, is fixed-parameter tractable on any class that
is nowhere dense and closed under induced subgraphs. This generalises known
results about the dominating set problem on -minor free classes, classes
with locally excluded minors and classes of graphs of bounded expansion. A
key feature of our proof is that it is based simply on the fact that these
graph classes are uniformly quasi-wide, and does not rely on a structural
decomposition. Our result also establishes that the distance-
dominating-set problem is FPT on classes of bounded expansion, answering a
question of Ne{v s}et{v r}il and Ossona de Mendez
Recommended from our members
Graph Theory
This workshop focused on recent developments in graph theory. These included in particular recent breakthroughs on nowhere-zero flows in graphs, width parameters, applications of graph sparsity in algorithms, and matroid structure results