761 research outputs found
The growth rate over trees of any family of set defined by a monadic second order formula is semi-computable
Monadic second order logic can be used to express many classical notions of
sets of vertices of a graph as for instance: dominating sets, induced
matchings, perfect codes, independent sets or irredundant sets. Bounds on the
number of sets of any such family of sets are interesting from a combinatorial
point of view and have algorithmic applications. Many such bounds on different
families of sets over different classes of graphs are already provided in the
literature. In particular, Rote recently showed that the number of minimal
dominating sets in trees of order is at most and that
this bound is asymptotically sharp up to a multiplicative constant. We build on
his work to show that what he did for minimal dominating sets can be done for
any family of sets definable by a monadic second order formula.
We first show that, for any monadic second order formula over graphs that
characterizes a given kind of subset of its vertices, the maximal number of
such sets in a tree can be expressed as the \textit{growth rate of a bilinear
system}. This mostly relies on well known links between monadic second order
logic over trees and tree automata and basic tree automata manipulations. Then
we show that this "growth rate" of a bilinear system can be approximated from
above.We then use our implementation of this result to provide bounds on the
number of independent dominating sets, total perfect dominating sets, induced
matchings, maximal induced matchings, minimal perfect dominating sets, perfect
codes and maximal irredundant sets on trees. We also solve a question from D.
Y. Kang et al. regarding -matchings and improve a bound from G\'orska and
Skupie\'n on the number of maximal matchings on trees. Remark that this
approach is easily generalizable to graphs of bounded tree width or clique
width (or any similar class of graphs where tree automata are meaningful)
Exploiting -Closure in Kernelization Algorithms for Graph Problems
A graph is c-closed if every pair of vertices with at least c common
neighbors is adjacent. The c-closure of a graph G is the smallest number such
that G is c-closed. Fox et al. [ICALP '18] defined c-closure and investigated
it in the context of clique enumeration. We show that c-closure can be applied
in kernelization algorithms for several classic graph problems. We show that
Dominating Set admits a kernel of size k^O(c), that Induced Matching admits a
kernel with O(c^7*k^8) vertices, and that Irredundant Set admits a kernel with
O(c^(5/2)*k^3) vertices. Our kernelization exploits the fact that c-closed
graphs have polynomially-bounded Ramsey numbers, as we show
On the approximability of the maximum induced matching problem
In this paper we consider the approximability of the maximum induced matching problem (MIM). We give an approximation algorithm with asymptotic performance ratio <i>d</i>-1 for MIM in <i>d</i>-regular graphs, for each <i>d</i>≥3. We also prove that MIM is APX-complete in <i>d</i>-regular graphs, for each <i>d</i>≥3
b-coloring is NP-hard on co-bipartite graphs and polytime solvable on tree-cographs
A b-coloring of a graph is a proper coloring such that every color class
contains a vertex that is adjacent to all other color classes. The b-chromatic
number of a graph G, denoted by \chi_b(G), is the maximum number t such that G
admits a b-coloring with t colors. A graph G is called b-continuous if it
admits a b-coloring with t colors, for every t = \chi(G),\ldots,\chi_b(G), and
b-monotonic if \chi_b(H_1) \geq \chi_b(H_2) for every induced subgraph H_1 of
G, and every induced subgraph H_2 of H_1.
We investigate the b-chromatic number of graphs with stability number two.
These are exactly the complements of triangle-free graphs, thus including all
complements of bipartite graphs. The main results of this work are the
following:
- We characterize the b-colorings of a graph with stability number two in
terms of matchings with no augmenting paths of length one or three. We derive
that graphs with stability number two are b-continuous and b-monotonic.
- We prove that it is NP-complete to decide whether the b-chromatic number of
co-bipartite graph is at most a given threshold.
- We describe a polynomial time dynamic programming algorithm to compute the
b-chromatic number of co-trees.
- Extending several previous results, we show that there is a polynomial time
dynamic programming algorithm for computing the b-chromatic number of
tree-cographs. Moreover, we show that tree-cographs are b-continuous and
b-monotonic
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