256 research outputs found
Efficient domination and polarity
The thesis considers the following graph problems:
Efficient (Edge) Domination seeks for an independent vertex (edge) subset D such that all other vertices (edges) have exactly one neighbor in D. Polarity asks for a vertex subset that induces a complete multipartite graph and that contains a vertex of every induced P_3. Monopolarity is the special case of Polarity where the wanted vertex subset has to be independent. These problems are NP-complete in general, but efficiently solvable on various graph classes.
The thesis sharpens known NP-completeness results and presents new solvable cases
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
Graph Isomorphism for unit square graphs
In the past decades for more and more graph classes the Graph Isomorphism
Problem was shown to be solvable in polynomial time. An interesting family of
graph classes arises from intersection graphs of geometric objects. In this
work we show that the Graph Isomorphism Problem for unit square graphs,
intersection graphs of axis-parallel unit squares in the plane, can be solved
in polynomial time. Since the recognition problem for this class of graphs is
NP-hard we can not rely on standard techniques for geometric graphs based on
constructing a canonical realization. Instead, we develop new techniques which
combine structural insights into the class of unit square graphs with
understanding of the automorphism group of such graphs. For the latter we
introduce a generalization of bounded degree graphs which is used to capture
the main structure of unit square graphs. Using group theoretic algorithms we
obtain sufficient information to solve the isomorphism problem for unit square
graphs.Comment: 31 pages, 6 figure
Characterizing the easy-to-find subgraphs from the viewpoint of polynomial-time algorithms, kernels, and Turing kernels
We study two fundamental problems related to finding subgraphs: (1) given
graphs G and H, Subgraph Test asks if H is isomorphic to a subgraph of G, (2)
given graphs G, H, and an integer t, Packing asks if G contains t
vertex-disjoint subgraphs isomorphic to H. For every graph class F, let
F-Subgraph Test and F-Packing be the special cases of the two problems where H
is restricted to be in F. Our goal is to study which classes F make the two
problems tractable in one of the following senses:
* (randomized) polynomial-time solvable,
* admits a polynomial (many-one) kernel, or
* admits a polynomial Turing kernel (that is, has an adaptive polynomial-time
procedure that reduces the problem to a polynomial number of instances, each of
which has size bounded polynomially by the size of the solution).
We identify a simple combinatorial property such that if a hereditary class F
has this property, then F-Packing admits a polynomial kernel, and has no
polynomial (many-one) kernel otherwise, unless the polynomial hierarchy
collapses. Furthermore, if F does not have this property, then F-Packing is
either WK[1]-hard, W[1]-hard, or Long Path-hard, giving evidence that it does
not admit polynomial Turing kernels either.
For F-Subgraph Test, we show that if every graph of a hereditary class F
satisfies the property that it is possible to delete a bounded number of
vertices such that every remaining component has size at most two, then
F-Subgraph Test is solvable in randomized polynomial time and it is NP-hard
otherwise. We introduce a combinatorial property called (a,b,c,d)-splittability
and show that if every graph in a hereditary class F has this property, then
F-Subgraph Test admits a polynomial Turing kernel and it is WK[1]-hard,
W[1]-hard, or Long Path-hard, otherwise.Comment: 69 pages, extended abstract to appear in the proceedings of SODA 201
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