251 research outputs found
Contraction Obstructions for Connected Graph Searching
We consider the connected variant of the classic mixed search game where, in
each search step, cleaned edges form a connected subgraph. We consider graph
classes with bounded connected (and monotone) mixed search number and we deal
with the question whether the obstruction set, with respect of the contraction
partial ordering, for those classes is finite. In general, there is no
guarantee that those sets are finite, as graphs are not well quasi ordered
under the contraction partial ordering relation.
In this paper we provide the obstruction set for , where is the
number of searchers we are allowed to use. This set is finite, it consists of
177 graphs and completely characterises the graphs with connected (and
monotone) mixed search number at most 2. Our proof reveals that the "sense of
direction" of an optimal search searching is important for connected search
which is in contrast to the unconnected original case. We also give a double
exponential lower bound on the size of the obstruction set for the classes
where this set is finite
Hyperopic Cops and Robbers
We introduce a new variant of the game of Cops and Robbers played on graphs,
where the robber is invisible unless outside the neighbor set of a cop. The
hyperopic cop number is the corresponding analogue of the cop number, and we
investigate bounds and other properties of this parameter. We characterize the
cop-win graphs for this variant, along with graphs with the largest possible
hyperopic cop number. We analyze the cases of graphs with diameter 2 or at
least 3, focusing on when the hyperopic cop number is at most one greater than
the cop number. We show that for planar graphs, as with the usual cop number,
the hyperopic cop number is at most 3. The hyperopic cop number is considered
for countable graphs, and it is shown that for connected chains of graphs, the
hyperopic cop density can be any real number in $[0,1/2].
Planar Induced Subgraphs of Sparse Graphs
We show that every graph has an induced pseudoforest of at least
vertices, an induced partial 2-tree of at least vertices, and an
induced planar subgraph of at least vertices. These results are
constructive, implying linear-time algorithms to find the respective induced
subgraphs. We also show that the size of the largest -minor-free graph in
a given graph can sometimes be at most .Comment: Accepted by Graph Drawing 2014. To appear in Journal of Graph
Algorithms and Application
Centroidal localization game
One important problem in a network is to locate an (invisible) moving entity
by using distance-detectors placed at strategical locations. For instance, the
metric dimension of a graph is the minimum number of detectors placed
in some vertices such that the vector
of the distances between the detectors and the entity's location
allows to uniquely determine . In a more realistic setting, instead
of getting the exact distance information, given devices placed in
, we get only relative distances between the entity's
location and the devices (for every , it is provided
whether , , or to ). The centroidal dimension of a
graph is the minimum number of devices required to locate the entity in
this setting.
We consider the natural generalization of the latter problem, where vertices
may be probed sequentially until the moving entity is located. At every turn, a
set of vertices is probed and then the relative distances
between the vertices and the current location of the entity are
given. If not located, the moving entity may move along one edge. Let be the minimum such that the entity is eventually located, whatever it
does, in the graph .
We prove that for every tree and give an upper bound
on in cartesian product of graphs and . Our main
result is that for any outerplanar graph . We then prove
that is bounded by the pathwidth of plus 1 and that the
optimization problem of determining is NP-hard in general graphs.
Finally, we show that approximating (up to any constant distance) the entity's
location in the Euclidean plane requires at most two vertices per turn
Classical Ising model test for quantum circuits
We exploit a recently constructed mapping between quantum circuits and graphs
in order to prove that circuits corresponding to certain planar graphs can be
efficiently simulated classically. The proof uses an expression for the Ising
model partition function in terms of quadratically signed weight enumerators
(QWGTs), which are polynomials that arise naturally in an expansion of quantum
circuits in terms of rotations involving Pauli matrices. We combine this
expression with a known efficient classical algorithm for the Ising partition
function of any planar graph in the absence of an external magnetic field, and
the Robertson-Seymour theorem from graph theory. We give as an example a set of
quantum circuits with a small number of non-nearest neighbor gates which admit
an efficient classical simulation.Comment: 17 pages, 2 figures. v2: main result strengthened by removing
oracular settin
A Connected Version of the Graph Coloring Game
The graph coloring game is a two-player game in which, given a graph G and a set of k colors, the two players, Alice and Bob, take turns coloring properly an uncolored vertex of G, Alice having the first move. Alice wins the game if and only if all the vertices of G are eventually colored. The game chromatic number of a graph G is then defined as the smallest integer k for which Alice has a winning strategy when playing the graph coloring game on G with k colors. In this paper, we introduce and study a new version of the graph coloring game by requiring that, after each player's turn, the subgraph induced by the set of colored vertices is connected. The connected game chromatic number of a graph G is then the smallest integer k for which Alice has a winning strategy when playing the connected graph coloring game on G with k colors. We prove that the connected game chromatic number of every outerplanar graph is at most 5 and that there exist outerplanar graphs with connected game chromatic number 4. Moreover, we prove that for every integer k ≥ 3, there exist bipartite graphs on which Bob wins the connected coloring game with k colors, while Alice wins the connected coloring game with two colors on every bipartite graph
Approximately Counting Embeddings into Random Graphs
Let H be a graph, and let C_H(G) be the number of (subgraph isomorphic)
copies of H contained in a graph G. We investigate the fundamental problem of
estimating C_H(G). Previous results cover only a few specific instances of this
general problem, for example, the case when H has degree at most one
(monomer-dimer problem). In this paper, we present the first general subcase of
the subgraph isomorphism counting problem which is almost always efficiently
approximable. The results rely on a new graph decomposition technique.
Informally, the decomposition is a labeling of the vertices such that every
edge is between vertices with different labels and for every vertex all
neighbors with a higher label have identical labels. The labeling implicitly
generates a sequence of bipartite graphs which permits us to break the problem
of counting embeddings of large subgraphs into that of counting embeddings of
small subgraphs. Using this method, we present a simple randomized algorithm
for the counting problem. For all decomposable graphs H and all graphs G, the
algorithm is an unbiased estimator. Furthermore, for all graphs H having a
decomposition where each of the bipartite graphs generated is small and almost
all graphs G, the algorithm is a fully polynomial randomized approximation
scheme.
We show that the graph classes of H for which we obtain a fully polynomial
randomized approximation scheme for almost all G includes graphs of degree at
most two, bounded-degree forests, bounded-length grid graphs, subdivision of
bounded-degree graphs, and major subclasses of outerplanar graphs,
series-parallel graphs and planar graphs, whereas unbounded-length grid graphs
are excluded.Comment: Earlier version appeared in Random 2008. Fixed an typo in Definition
3.
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