66,133 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)
A characterization of trees with equal 2-domination and 2-independence numbers
A set of vertices in a graph is a -dominating set if every vertex
of not in is adjacent to at least two vertices in , and is a
-independent set if every vertex in is adjacent to at most one vertex of
. The -domination number is the minimum cardinality of a
-dominating set in , and the -independence number is the
maximum cardinality of a -independent set in . Chellali and Meddah [{\it
Trees with equal -domination and -independence numbers,} Discussiones
Mathematicae Graph Theory 32 (2012), 263--270] provided a constructive
characterization of trees with equal -domination and -independence
numbers. Their characterization is in terms of global properties of a tree, and
involves properties of minimum -dominating and maximum -independent sets
in the tree at each stage of the construction. We provide a constructive
characterization that relies only on local properties of the tree at each stage
of the construction.Comment: 17 pages, 4 figure
Bounds for the Number of Independent and Dominating Sets in Trees
In this work, we investigate bounds on the number of independent sets in a graph and its complement, along with the corresponding question for number of dominating sets. Nordhaus and Gaddum gave bounds on χ(G)+χ(G) and χ(G) χ(G), where G is any graph on n vertices and χ(G) is the chromatic number of G. Nordhaus-Gaddum- type inequalities have been studied for many other graph invariants. In this work, we concentrate on i(G), the number of independent sets in G, and ∂(G), the number of dominating sets in G. We focus our attention on Nordhaus-Gaddum-type inequalities over trees on a fixed number of vertices. In particular, we give sharp upper and lower bounds on i(T )+ i(T ) where T is a tree on n vertices, improving bounds and proofs of Hu and Wei. We also give upper and lower bounds on i(G) + i(G) where G is a unicyclic graph on n vertices, again improving a result of Hu and Wei. Lastly, we investigate ∂(T )+ ∂(T ) where T is a tree on n vertices. We use a result of Wagner to give a lower bound and make a conjecture about an upper bound
Tight Complexity Bounds for Counting Generalized Dominating Sets in Bounded-Treewidth Graphs
We investigate how efficiently a well-studied family of domination-type
problems can be solved on bounded-treewidth graphs. For sets of
non-negative integers, a -set of a graph is a set of
vertices such that for every , and for every . The problem of finding a
-set (of a certain size) unifies standard problems such as
Independent Set, Dominating Set, Independent Dominating Set, and many others.
For all pairs of finite or cofinite sets , we determine (under
standard complexity assumptions) the best possible value such
that there is an algorithm that counts -sets in time
(if a tree decomposition of width
is given in the input). For example, for the Exact Independent
Dominating Set problem (also known as Perfect Code) corresponding to
and , we improve the
algorithm of [van Rooij, 2020] to .
Despite the unusually delicate definition of , we show that
our algorithms are most likely optimal, i.e., for any pair of
finite or cofinite sets where the problem is non-trivial, and any
, a -algorithm counting the number of -sets would violate
the Counting Strong Exponential-Time Hypothesis (#SETH). For finite sets
and , our lower bounds also extend to the decision version,
showing that our algorithms are optimal in this setting as well. In contrast,
for many cofinite sets, we show that further significant improvements for the
decision and optimization versions are possible using the technique of
representative sets
Independent transversal total domination versus total domination in trees
A subset of vertices in a graph G is a total dominating set if every vertex in G is adjacent to at least one vertex in this subset. The total domination number of G is the minimum cardinality of any total dominating set in G and is denoted by gamma(t)(G). A total dominating set of G having nonempty intersection with all the independent sets of maximum cardinality in G is an independent transversal total dominating set. The minimum cardinality of any independent transversal total dominating set is denoted by gamma(u) (G). Based on the fact that for any tree T, gamma(t) (T) <= gamma(u) (T) <= gamma(t) (T) + 1, in this work we give several relationship(s) between gamma(u) (T) and gamma(t) (T) for trees T which are leading to classify the trees which are satisfying the equality in these bound
In the complement of a dominating set
A set D of vertices of a graph G=(V,E) is a dominating set, if every vertex
of D\V has at least one neighbor that belongs to D. The disjoint domination
number of a graph G is the minimum cardinality of two disjoint dominating
sets of G. We prove upper bounds for the disjoint domination number for
graphs of minimum degree at least 2, for graphs of large minimum degree and
for cubic graphs.A set T of vertices of a graph G=(V,E) is a total
dominating set, if every vertex of G has at least one neighbor that belongs
to T. We characterize graphs of minimum degree 2 without induced 5-cycles
and graphs of minimum degree at least 3 that have a dominating set, a total
dominating set, and a non-empty vertex set that are disjoint.A set I of
vertices of a graph G=(V,E) is an independent set, if all vertices in I are
not adjacent in G. We give a constructive characterization of trees that
have a maximum independent set and a minimum dominating set that are
disjoint and we show that the corresponding decision problem is NP-hard for
general graphs. Additionally, we prove several structural and hardness
results concerning pairs of disjoint sets in graphs which are dominating,
independent, or both. Furthermore, we prove lower bounds for the maximum
cardinality of an independent set of graphs with specifed odd girth and
small average degree.A connected graph G has spanning tree congestion at
most s, if G has a spanning tree T such that for every edge e of T the edge
cut defined in G by the vertex sets of the two components of T-e contains
at most s edges. We prove that every connected graph of order n has
spanning tree congestion at most n^(3/2) and we show that the corresponding
decision problem is NP-hard
Reconfiguration of Dominating Sets
We explore a reconfiguration version of the dominating set problem, where a
dominating set in a graph is a set of vertices such that each vertex is
either in or has a neighbour in . In a reconfiguration problem, the goal
is to determine whether there exists a sequence of feasible solutions
connecting given feasible solutions and such that each pair of
consecutive solutions is adjacent according to a specified adjacency relation.
Two dominating sets are adjacent if one can be formed from the other by the
addition or deletion of a single vertex.
For various values of , we consider properties of , the graph
consisting of a vertex for each dominating set of size at most and edges
specified by the adjacency relation. Addressing an open question posed by Haas
and Seyffarth, we demonstrate that is not necessarily
connected, for the maximum cardinality of a minimal dominating set
in . The result holds even when graphs are constrained to be planar, of
bounded tree-width, or -partite for . Moreover, we construct an
infinite family of graphs such that has exponential
diameter, for the minimum size of a dominating set. On the positive
side, we show that is connected and of linear diameter for any
graph on vertices having at least independent edges.Comment: 12 pages, 4 figure
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