28 research outputs found
Total Domishold Graphs: a Generalization of Threshold Graphs, with Connections to Threshold Hypergraphs
A total dominating set in a graph is a set of vertices such that every vertex
of the graph has a neighbor in the set. We introduce and study graphs that
admit non-negative real weights associated to their vertices such that a set of
vertices is a total dominating set if and only if the sum of the corresponding
weights exceeds a certain threshold. We show that these graphs, which we call
total domishold graphs, form a non-hereditary class of graphs properly
containing the classes of threshold graphs and the complements of domishold
graphs, and are closely related to threshold Boolean functions and threshold
hypergraphs. We present a polynomial time recognition algorithm of total
domishold graphs, and characterize graphs in which the above property holds in
a hereditary sense. Our characterization is obtained by studying a new family
of hypergraphs, defined similarly as the Sperner hypergraphs, which may be of
independent interest.Comment: 19 pages, 1 figur
Shortest paths between shortest paths and independent sets
We study problems of reconfiguration of shortest paths in graphs. We prove
that the shortest reconfiguration sequence can be exponential in the size of
the graph and that it is NP-hard to compute the shortest reconfiguration
sequence even when we know that the sequence has polynomial length. Moreover,
we also study reconfiguration of independent sets in three different models and
analyze relationships between these models, observing that shortest path
reconfiguration is a special case of independent set reconfiguration in perfect
graphs, under any of the three models. Finally, we give polynomial results for
restricted classes of graphs (even-hole-free and -free graphs)
Domination parameters with number 2: interrelations and algorithmic consequences
In this paper, we study the most basic domination invariants in graphs, in
which number 2 is intrinsic part of their definitions. We classify them upon
three criteria, two of which give the following previously studied invariants:
the weak -domination number, , the -domination number,
, the -domination number, , the double
domination number, , the total -domination number,
, and the total double domination number, , where is a graph in which a corresponding invariant is well
defined. The third criterion yields rainbow versions of the mentioned six
parameters, one of which has already been well studied, and three other give
new interesting parameters. Together with a special, extensively studied Roman
domination, , and two classical parameters, the domination number,
, and the total domination number, , we consider 13
domination invariants in graphs . In the main result of the paper we present
sharp upper and lower bounds of each of the invariants in terms of every other
invariant, large majority of which are new results proven in this paper. As a
consequence of the main theorem we obtain some complexity results for the
studied invariants, in particular regarding the existence of approximation
algorithms and inapproximability bounds.Comment: 45 pages, 4 tables, 7 figure
Complexity Results for Equistable Graphs and Related Classes
The class of equistable graphs is defined by the existence of a cost structure on the vertices such that the maximal stable sets are characterized by their costs. This graph class, not contained in any nontrivial hereditary class, has so far been studied mostly from a structural point of view; characterizations and polynomial time recognition algorithms have been obtained for special cases.
We focus on complexity issues for equistable graphs and related classes. We describe a simple pseudo-polynomial-time dynamic programming algorithm to solve the maximum weight stable set problem along with the weighted independent domination problem in some classes of graphs, including equistable graphs. Our results are obtained within the wider context of Boolean optimization; corresponding hardness results are also provided. More specifically, we show that the above problems are APX-hard for equistable graphs and that it is co-NP-complete to determine whether a given cost function on the vertices of a graph defines an equistable cost structure of that graph.Germany. Federal Ministry of Education and ResearchAlexander von Humboldt-Stiftung (Sofja Kovalevskaja Award 2004
Graphs of Separability at Most Two: Structural Characterizations and Their Consequences
Graphs of separability at most k are defined as graphs in which every two non-adjacent vertices are separated by a set of at most k other vertices. For k 08 {0,1}, the only connected graphs of separability at most k are complete graphs and block graphs, respectively. For k 65 3, graphs of separability at most k form a rich class of graphs containing all graphs of maximum degree k. Graphs of separability at most 2 generalize complete graphs, cycles and trees. We prove several characterizations of graphs of separability at most 2 and examine some of their consequences
On the maximum independent set problem in subclasses of planar graphs
The maximum independent set problem is known to be NP-hard in the class of planar graphs. In the present paper, we study its complexity in hereditary subclasses of planar graphs. In particular, by combining various techniques, we show that the problem is polynomially solvable in the class of S1,2,k-free planar graphs, generalizing several previously known results. S1,2,k is the graph consisting of three induced paths of lengths 1, 2 and k, with a common initial vertex
Tree-width and optimization in bounded degree graphs
It is well known that boundedness of tree-width implies polynomial-time solvability of many algorithmic graph problems. The converse statement is generally not true, i.e., polynomial-time solvability does not necessarily imply boundedness of tree-width. However, in graphs of bounded vertex degree, for some problems, the two concepts behave in a more consistent way. In the present paper, we study this phenomenon with respect to three important graph problems - dominating set, independent dominating set and induced matching - and obtain several results toward revealing the equivalency between boundedness of the tree-width and polynomial-time solvability of these problems in bounded degree graphs