314 research outputs found
Maximum Weight Independent Sets in Odd-Hole-Free Graphs Without Dart or Without Bull
The Maximum Weight Independent Set (MWIS) Problem on graphs with vertex
weights asks for a set of pairwise nonadjacent vertices of maximum total
weight. Being one of the most investigated and most important problems on
graphs, it is well known to be NP-complete and hard to approximate. The
complexity of MWIS is open for hole-free graphs (i.e., graphs without induced
subgraphs isomorphic to a chordless cycle of length at least five). By applying
clique separator decomposition as well as modular decomposition, we obtain
polynomial time solutions of MWIS for odd-hole- and dart-free graphs as well as
for odd-hole- and bull-free graphs (dart and bull have five vertices, say
, and dart has edges , while bull has edges
). If the graphs are hole-free instead of odd-hole-free then
stronger structural results and better time bounds are obtained
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
List Locally Surjective Homomorphisms in Hereditary Graph Classes
A locally surjective homomorphism from a graph G to a graph H is an edge-preserving mapping from V(G) to V(H) that is surjective in the neighborhood of each vertex in G. In the list locally surjective homomorphism problem, denoted by LLSHom(H), the graph H is fixed and the instance consists of a graph G whose every vertex is equipped with a subset of V(H), called list. We ask for the existence of a locally surjective homomorphism from G to H, where every vertex of G is mapped to a vertex from its list. In this paper, we study the complexity of the LLSHom(H) problem in F-free graphs, i.e., graphs that exclude a fixed graph F as an induced subgraph. We aim to understand for which pairs (H,F) the problem can be solved in subexponential time.
We show that for all graphs H, for which the problem is NP-hard in general graphs, it cannot be solved in subexponential time in F-free graphs for F being a bounded-degree forest, unless the ETH fails. The initial study reveals that a natural subfamily of bounded-degree forests F, that might lead to some tractability results, is the family ? consisting of forests whose every component has at most three leaves. In this case, we exhibit the following dichotomy theorem: besides the cases that are polynomial-time solvable in general graphs, the graphs H ? {P?,C?} are the only connected ones that allow for a subexponential-time algorithm in F-free graphs for every F ? ? (unless the ETH fails)
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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