A Boundary Property for Upper Domination

An upper dominating set in a graph is a minimal (with respect to set inclusion) dominating set of maximum cardinality. The problem of finding an upper dominating set is generally NP-hard, but can be solved in polynomial time in some restricted graph classes, such as P4-free graphs or 2K2-free graphs. For classes defined by finitely many forbidden induced subgraphs, the boundary separating difficult instances of the problem from polynomially solvable ones consists of the so called boundary classes. However, none of such classes has been identified so far for the upper dominating set problem. In the present paper, we discover the first boundary class for this problem

Upper Domination: Towards a Dichotomy Through Boundary Properties

Anupper dominating set in a graph is a minimal dominating set of maximum cardinality. The problem of finding an upper dominating set is generally NP-hard.We study the complexity of this problem in finitely defined classes of graphs and conjecture that the problem admits a complexity dichotomy in this family. A helpful tool to study the complexity of an algorithmic problem is the notion of boundary classes. However, none of such classes has been identified so far for the upper dominating set problem. We discover the first boundary class for this problem and prove the dichotomy for monogenic classes

Critical properties and complexity measures of read-once Boolean functions

In this paper, we define a quasi-order on the set of read-once Boolean functions and show that this is a well-quasi-order. This implies that every parameter measuring complexity of the functions can be characterized by a finite set of minimal subclasses of read-once functions, where this parameter is unbounded. We focus on two parameters related to certificate complexity and characterize each of them in the terminology of minimal classes

Upper Domination: Towards a Dichotomy Through Boundary Properties

An upper dominating set in a graph is a minimal dominating set of maximum cardinality. The problem of finding an upper dominating set is generally NP-hard. We study the complexity of this problem in finitely defined classes of graphs and conjecture that the problem admits a complexity dichotomy in this family. A helpful tool to study the complexity of an algorithmic problem is the notion of boundary classes. However, none of such classes has been identified so far for the upper dominating set problem. We discover the first boundary class for this problem and prove the dichotomy for monogenic classes

Critical properties of graphs of bounded clique-width

A graph property is a set of graphs closed under isomorphism. Clique-width is a graph parameter which is important in theoretical computer science because many algorithmic problems that are generally NP-hard admit polynomial-time solutions when restricted to graphs of bounded clique-width. Over the last few years, many properties of graphs have been shown to be of bounded clique-width; for many others, it has been shown that the clique-width is unbounded. The goal of the present paper is to tighten the gap between properties of bounded and unbounded clique-width. To this end, we identify new necessary and sufficient conditions for clique-width to be bounded