17,195 research outputs found
Spectral bounds for the k-regular induced subgraph problem
Many optimization problems on graphs are reduced to the determination of a subset of vertices of maximum cardinality which induces a -regular subgraph. For example, a maximum independent set, a maximum induced matching and a maximum clique is a maximum cardinality -regular, -regular and -regular induced subgraph,
respectively, were denotes the clique number of the graph . The determination of the order of a -regular induced subgraph of highest order is in general an NP-hard problem. This paper is devoted to the study of spectral upper bounds on the order of these subgraphs which are determined in polynomial time and in many cases are good approximations of the respective optimal solutions. The introduced upper bounds are deduced based on adjacency,
Laplacian and signless Laplacian spectra. Some analytical comparisons between them are presented. Finally, all of the studied upper bounds are tested and compared through several computational experiments
Majorantes para a ordem de subgrafos induzidos k-regulares
Doutoramento em MatemáticaMuitos dos problemas de otimização em grafos reduzem-se à determinação
de um subconjunto de vértices de cardinalidade máxima que induza
um subgrafo k-regular. Uma vez que a determinação da ordem
de um subgrafo induzido k-regular de maior ordem é, em geral, um
problema NP-difícil, são deduzidos novos majorantes, a determinar em
tempo polinomial, que em muitos casos constituam boas aproximações
das respetivas soluções ótimas. Introduzem-se majorantes espetrais
usando uma abordagem baseada em técnicas de programação convexa
e estabelecem-se condições necessárias e suficientes para que sejam
atingidos. Adicionalmente, introduzem-se majorantes baseados no espetro
das matrizes de adjacência, laplaciana e laplaciana sem sinal. É
ainda apresentado um algoritmo não polinomial para a determinação de
umsubconjunto de vértices de umgrafo que induz umsubgrafo k-regular
de ordem máxima para uma classe particular de grafos. Finalmente,
faz-se um estudo computacional comparativo com vários majorantes e
apresentam-se algumas conclusões.Many optimization problems on graphs are reduced to the determination
of a subset of vertices of maximum cardinality inducing a k-regular
subgraph. Since the determination of the order of a k-regular induced
subgraph of highest order is in general a NP-hard problem, new upper
bounds, determined in polynomial time which in many cases are good
approximations of the respective optimal solutions are deduced. Using
convex programming techniques, spectral upper boundswere introduced
jointly with necessary and sufficient conditions for those upper bounds
be achieved. Additionally, upper bounds based on adjacency, Laplacian
and signless Laplacian spectrum are introduced. Furthermore, a nonpolynomial
time algorithm for determining a subset of vertices of a graph
which induces a maximum k-regular induced subgraph for a particular
class is presented. Finally, a comparative computational study is provided
jointly with a few conclusions
On giant components and treewidth in the layers model
Given an undirected -vertex graph and an integer , let
denote the random vertex induced subgraph of generated by ordering
according to a random permutation and including in those
vertices with at most of their neighbors preceding them in this order.
The distribution of subgraphs sampled in this manner is called the \emph{layers
model with parameter} . The layers model has found applications in studying
-degenerate subgraphs, the design of algorithms for the maximum
independent set problem, and in bootstrap percolation.
In the current work we expand the study of structural properties of the
layers model.
We prove that there are -regular graphs for which with high
probability has a connected component of size . Moreover,
this connected component has treewidth . This lower bound on the
treewidth extends to many other random graph models. In contrast, is
known to be a forest (hence of treewidth~1), and we establish that if is of
bounded degree then with high probability the largest connected component in
is of size . We also consider the infinite two-dimensional
grid, for which we prove that the first four layers contain a unique infinite
connected component with probability
A complete solution to the infinite Oberwolfach problem
Let be a -regular graph of order . The Oberwolfach problem,
, asks for a -factorization of the complete graph on vertices in
which each -factor is isomorphic to . In this paper, we give a complete
solution to the Oberwolfach problem over infinite complete graphs, proving the
existence of solutions that are regular under the action of a given involution
free group . We will also consider the same problem in the more general
contest of graphs that are spanning subgraphs of an infinite complete graph
and we provide a solution when is locally finite. Moreover, we
characterize the infinite subgraphs of such that there exists a
solution to containing a solution to
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
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
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