195 research outputs found
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
On Minimum Maximal Distance-k Matchings
We study the computational complexity of several problems connected with
finding a maximal distance- matching of minimum cardinality or minimum
weight in a given graph. We introduce the class of -equimatchable graphs
which is an edge analogue of -equipackable graphs. We prove that the
recognition of -equimatchable graphs is co-NP-complete for any fixed . We provide a simple characterization for the class of strongly chordal
graphs with equal -packing and -domination numbers. We also prove that
for any fixed integer the problem of finding a minimum weight
maximal distance- matching and the problem of finding a minimum weight
-independent dominating set cannot be approximated in polynomial
time in chordal graphs within a factor of unless
, where is a fixed constant (thereby
improving the NP-hardness result of Chang for the independent domination case).
Finally, we show the NP-hardness of the minimum maximal induced matching and
independent dominating set problems in large-girth planar graphs.Comment: 15 pages, 4 figure
Some results on triangle partitions
We show that there exist efficient algorithms for the triangle packing
problem in colored permutation graphs, complete multipartite graphs,
distance-hereditary graphs, k-modular permutation graphs and complements of
k-partite graphs (when k is fixed). We show that there is an efficient
algorithm for C_4-packing on bipartite permutation graphs and we show that
C_4-packing on bipartite graphs is NP-complete. We characterize the cobipartite
graphs that have a triangle partition
Clique-Stable Set separation in perfect graphs with no balanced skew-partitions
Inspired by a question of Yannakakis on the Vertex Packing polytope of
perfect graphs, we study the Clique-Stable Set Separation in a non-hereditary
subclass of perfect graphs. A cut (B,W) of G (a bipartition of V(G)) separates
a clique K and a stable set S if and . A
Clique-Stable Set Separator is a family of cuts such that for every clique K,
and for every stable set S disjoint from K, there exists a cut in the family
that separates K and S. Given a class of graphs, the question is to know
whether every graph of the class admits a Clique-Stable Set Separator
containing only polynomially many cuts. It is open for the class of all graphs,
and also for perfect graphs, which was Yannakakis' original question. Here we
investigate on perfect graphs with no balanced skew-partition; the balanced
skew-partition was introduced in the proof of the Strong Perfect Graph Theorem.
Recently, Chudnovsky, Trotignon, Trunck and Vuskovic proved that forbidding
this unfriendly decomposition permits to recursively decompose Berge graphs
using 2-join and complement 2-join until reaching a basic graph, and they found
an efficient combinatorial algorithm to color those graphs. We apply their
decomposition result to prove that perfect graphs with no balanced
skew-partition admit a quadratic-size Clique-Stable Set Separator, by taking
advantage of the good behavior of 2-join with respect to this property. We then
generalize this result and prove that the Strong Erdos-Hajnal property holds in
this class, which means that every such graph has a linear-size biclique or
complement biclique. This property does not hold for all perfect graphs (Fox
2006), and moreover when the Strong Erdos-Hajnal property holds in a hereditary
class of graphs, then both the Erdos-Hajnal property and the polynomial
Clique-Stable Set Separation hold.Comment: arXiv admin note: text overlap with arXiv:1308.644
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