41 research outputs found
Every Property of Outerplanar Graphs is Testable
A D-disc around a vertex v of a graph G=(V,E) is the subgraph induced by all vertices of distance at most D from v. We show that the structure of an outerplanar graph on n vertices is determined, up to modification (insertion or deletion) of at most epsilon n edges, by a set of D-discs around the vertices, for D=D(epsilon) that is independent of the size of the graph. Such a result was already known for planar graphs (and any hyperfinite graph class), in the limited case of bounded degree graphs (that is, their maximum degree is bounded by some fixed constant, independent of |V|). We prove this result with no assumption on the degree of the graph.
A pure combinatorial consequence of this result is that two outerplanar graphs that share the same local views are close to be isomorphic.
We also obtain the following property testing results in the sparse graph model:
* graph isomorphism is testable for outerplanar graphs by poly(log n) queries.
* every graph property is testable for outerplanar graphs by poly(log n) queries.
We note that we can replace outerplanar graphs by a slightly more general family of k-edge-outerplanar graphs. The only previous general testing results, as above, where known for forests (Kusumoto and Yoshida), and for some power-law graphs that are extremely close to be bounded degree hyperfinite (by Ito)
Beyond Outerplanarity
We study straight-line drawings of graphs where the vertices are placed in
convex position in the plane, i.e., convex drawings. We consider two families
of graph classes with nice convex drawings: outer -planar graphs, where each
edge is crossed by at most other edges; and, outer -quasi-planar graphs
where no edges can mutually cross. We show that the outer -planar graphs
are -degenerate, and consequently that every
outer -planar graph can be -colored, and this
bound is tight. We further show that every outer -planar graph has a
balanced separator of size . This implies that every outer -planar
graph has treewidth . For fixed , these small balanced separators
allow us to obtain a simple quasi-polynomial time algorithm to test whether a
given graph is outer -planar, i.e., none of these recognition problems are
NP-complete unless ETH fails. For the outer -quasi-planar graphs we prove
that, unlike other beyond-planar graph classes, every edge-maximal -vertex
outer -quasi planar graph has the same number of edges, namely . We also construct planar 3-trees that are not outer
-quasi-planar. Finally, we restrict outer -planar and outer
-quasi-planar drawings to \emph{closed} drawings, where the vertex sequence
on the boundary is a cycle in the graph. For each , we express closed outer
-planarity and \emph{closed outer -quasi-planarity} in extended monadic
second-order logic. Thus, closed outer -planarity is linear-time testable by
Courcelle's Theorem.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Distributed Testing of Graph Isomorphism in the CONGEST Model
In this paper we study the problem of testing graph isomorphism (GI) in the
CONGEST distributed model. In this setting we test whether the distributive
network, , is isomorphic to which is given as an input to all the
nodes in the network, or alternatively, only to a single node.
We first consider the decision variant of the problem in which the algorithm
distinguishes and which are isomorphic from and which
are not isomorphic. We provide a randomized algorithm with rounds for
the setting in which is given only to a single node. We prove that for
this setting the number of rounds of any deterministic algorithm is
rounds, where denotes the number of nodes, which
implies a separation between the randomized and the deterministic complexities
of deciding GI.
We then consider the \emph{property testing} variant of the problem, where
the algorithm is only required to distinguish the case that and are
isomorphic from the case that and are \emph{far} from being
isomorphic (according to some predetermined distance measure). We show that
every algorithm requires rounds, where denotes the diameter of
the network. This lower bound holds even if all the nodes are given as an
input, and even if the message size is unbounded. We provide a randomized
algorithm with an almost matching round complexity of rounds that is suitable for dense graphs.
We also show that with the same number of rounds it is possible that each
node outputs its mapping according to a bijection which is an
\emph{approximated} isomorphism.
We conclude with simple simulation arguments that allow us to obtain
essentially tight algorithms with round complexity for special
families of sparse graphs
A Sublinear Tester for Outerplanarity (and Other Forbidden Minors) With One-Sided Error
We consider one-sided error property testing of -minor freeness
in bounded-degree graphs for any finite family of graphs that
contains a minor of , the -circus graph, or the -grid
for any . This includes, for instance, testing whether a graph
is outerplanar or a cactus graph. The query complexity of our algorithm in
terms of the number of vertices in the graph, , is . Czumaj et~al.\ showed that cycle-freeness and -minor
freeness can be tested with query complexity by using
random walks, and that testing -minor freeness for any that contains a
cycles requires queries. In contrast to these results, we
analyze the structure of the graph and show that either we can find a subgraph
of sublinear size that includes the forbidden minor , or we can find a pair
of disjoint subsets of vertices whose edge-cut is large, which induces an
-minor.Comment: extended to testing outerplanarity, full version of ICALP pape
A Quasi-Polynomial Time Partition Oracle for Graphs with an Excluded Minor
Motivated by the problem of testing planarity and related properties, we
study the problem of designing efficient {\em partition oracles}. A {\em
partition oracle} is a procedure that, given access to the incidence lists
representation of a bounded-degree graph and a parameter \eps,
when queried on a vertex , returns the part (subset of vertices) which
belongs to in a partition of all graph vertices. The partition should be
such that all parts are small, each part is connected, and if the graph has
certain properties, the total number of edges between parts is at most \eps
|V|. In this work we give a partition oracle for graphs with excluded minors
whose query complexity is quasi-polynomial in 1/\eps, thus improving on the
result of Hassidim et al. ({\em Proceedings of FOCS 2009}) who gave a partition
oracle with query complexity exponential in 1/\eps. This improvement implies
corresponding improvements in the complexity of testing planarity and other
properties that are characterized by excluded minors as well as sublinear-time
approximation algorithms that work under the promise that the graph has an
excluded minor.Comment: 13 pages, 1 figur
The Subgraph Testing Model
We initiate a study of testing properties of graphs that are presented as subgraphs of a fixed (or an explicitly given) graph. The tester is given free access to a base graph G=([n],E), and oracle access to a function f:E -> {0,1} that represents a subgraph of G. The tester is required to distinguish between subgraphs that posses a predetermined property and subgraphs that are far from possessing this property.
We focus on bounded-degree base graphs and on the relation between testing graph properties in the subgraph model and testing the same properties in the bounded-degree graph model. We identify cases in which testing is significantly easier in one model than in the other as well as cases in which testing has approximately the same complexity in both models. Our proofs are based on the design and analysis of efficient testers and on the establishment of query-complexity lower bounds
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On 3-Pushdown Graphs With Large Separators
For an integer s let ZS(n), the s-iterated logarithm function, be defined inductively: [O(n) = n, [8+1(n) = log2(l8(n)) for s 2:: o. We show that for every fixed s and all n large enough, there is an n-vertex 3-pushdown graph whose smallest separator contains at least n(n/[8(n)) vertices
Large induced subgraphs via triangulations and CMSO
We obtain an algorithmic meta-theorem for the following optimization problem.
Let \phi\ be a Counting Monadic Second Order Logic (CMSO) formula and t be an
integer. For a given graph G, the task is to maximize |X| subject to the
following: there is a set of vertices F of G, containing X, such that the
subgraph G[F] induced by F is of treewidth at most t, and structure (G[F],X)
models \phi.
Some special cases of this optimization problem are the following generic
examples. Each of these cases contains various problems as a special subcase:
1) "Maximum induced subgraph with at most l copies of cycles of length 0
modulo m", where for fixed nonnegative integers m and l, the task is to find a
maximum induced subgraph of a given graph with at most l vertex-disjoint cycles
of length 0 modulo m.
2) "Minimum \Gamma-deletion", where for a fixed finite set of graphs \Gamma\
containing a planar graph, the task is to find a maximum induced subgraph of a
given graph containing no graph from \Gamma\ as a minor.
3) "Independent \Pi-packing", where for a fixed finite set of connected
graphs \Pi, the task is to find an induced subgraph G[F] of a given graph G
with the maximum number of connected components, such that each connected
component of G[F] is isomorphic to some graph from \Pi.
We give an algorithm solving the optimization problem on an n-vertex graph G
in time O(#pmc n^{t+4} f(t,\phi)), where #pmc is the number of all potential
maximal cliques in G and f is a function depending of t and \phi\ only. We also
show how a similar running time can be obtained for the weighted version of the
problem. Pipelined with known bounds on the number of potential maximal
cliques, we deduce that our optimization problem can be solved in time
O(1.7347^n) for arbitrary graphs, and in polynomial time for graph classes with
polynomial number of minimal separators
An Efficient Partitioning Oracle for Bounded-Treewidth Graphs
Partitioning oracles were introduced by Hassidim et al. (FOCS 2009) as a
generic tool for constant-time algorithms. For any epsilon > 0, a partitioning
oracle provides query access to a fixed partition of the input bounded-degree
minor-free graph, in which every component has size poly(1/epsilon), and the
number of edges removed is at most epsilon*n, where n is the number of vertices
in the graph.
However, the oracle of Hassidimet al. makes an exponential number of queries
to the input graph to answer every query about the partition. In this paper, we
construct an efficient partitioning oracle for graphs with constant treewidth.
The oracle makes only O(poly(1/epsilon)) queries to the input graph to answer
each query about the partition.
Examples of bounded-treewidth graph classes include k-outerplanar graphs for
fixed k, series-parallel graphs, cactus graphs, and pseudoforests. Our oracle
yields poly(1/epsilon)-time property testing algorithms for membership in these
classes of graphs. Another application of the oracle is a poly(1/epsilon)-time
algorithm that approximates the maximum matching size, the minimum vertex cover
size, and the minimum dominating set size up to an additive epsilon*n in graphs
with bounded treewidth. Finally, the oracle can be used to test in
poly(1/epsilon) time whether the input bounded-treewidth graph is k-colorable
or perfect.Comment: Full version of a paper to appear in RANDOM 201