11 research outputs found
Сложность задачи о наибольшем независимом множестве в классе треугольных графов
Секция 10. Теоретическая информатикаРассматриваются вопросы вычислительной сложности и сложности аппро-кимации задачи о наибольшем независимом множестве в классе треугольных графов. Вводится новый теоретико-графовый параметр – верхнее окрестностное число независимости – и связанная с ним задача распознавания (задача о верхнем окрестностном независимом множестве). Показано, что значение этого параметра для треугольного графа равно числу независимости. Установлено, что обе рассматриваемые задачи являются NP-полными в классе треугольных графов и не аппроксимируются за полиномиальное время с точностью до произвольной константы K > 1 (при условии P ≠ NP). Найдены также некоторые полиномиально разрешимые случаи этих задач для треугольных графов
Equistarable graphs and counterexamples to three conjectures on equistable graphs
Equistable graphs are graphs admitting positive weights on vertices such that
a subset of vertices is a maximal stable set if and only if it is of total
weight . In , Mahadev et al.~introduced a subclass of equistable
graphs, called strongly equistable graphs, as graphs such that for every and every non-empty subset of vertices that is not a maximal stable set,
there exist positive vertex weights such that every maximal stable set is of
total weight and the total weight of does not equal . Mahadev et al.
conjectured that every equistable graph is strongly equistable. General
partition graphs are the intersection graphs of set systems over a finite
ground set such that every maximal stable set of the graph corresponds to a
partition of . In , Orlin proved that every general partition graph is
equistable, and conjectured that the converse holds as well.
Orlin's conjecture, if true, would imply the conjecture due to Mahadev,
Peled, and Sun. An intermediate conjecture, one that would follow from Orlin's
conjecture and would imply the conjecture by Mahadev, Peled, and Sun, was posed
by Miklavi\v{c} and Milani\v{c} in , and states that every equistable
graph has a clique intersecting all maximal stable sets. The above conjectures
have been verified for several graph classes. We introduce the notion of
equistarable graphs and based on it construct counterexamples to all three
conjectures within the class of complements of line graphs of triangle-free
graphs
ХАРАКТЕРИЗАЦИЯ 1-ТРЕУГОЛЬНЫХ ГРАФОВ
A graph is called 1-triangle if for each maximal independent set I, each edge of this graph with both end vertices not belonging to I forms exactly one triangle with a vertex from the set I. We have obtained a structural characterization of 1-triangle graphs which implies a polynomial time recognition algorithm for this class of graphs.Граф называется 1-треугольным, если для любого максимального независимого множества I этого графа каждое ребро графа, не инцидентное ни одной вершине из I, образует единственный треугольник с вершиной из множества I. В работе получена структурная характеризация класса 1-треугольных графов, которая влечёт полиномиальный алгоритм их распознавания
Decomposing 1-Sperner hypergraphs
A hypergraph is Sperner if no hyperedge contains another one. A Sperner
hypergraph is equilizable (resp., threshold) if the characteristic vectors of
its hyperedges are the (minimal) binary solutions to a linear equation (resp.,
inequality) with positive coefficients. These combinatorial notions have many
applications and are motivated by the theory of Boolean functions and integer
programming. We introduce in this paper the class of -Sperner hypergraphs,
defined by the property that for every two hyperedges the smallest of their two
set differences is of size one. We characterize this class of Sperner
hypergraphs by a decomposition theorem and derive several consequences from it.
In particular, we obtain bounds on the size of -Sperner hypergraphs and
their transversal hypergraphs, show that the characteristic vectors of the
hyperedges are linearly independent over the reals, and prove that -Sperner
hypergraphs are both threshold and equilizable. The study of -Sperner
hypergraphs is motivated also by their applications in graph theory, which we
present in a companion paper
Strong cliques and equistability of EPT graphs
In this paper, we characterize the equistable graphs within the class of EPT graphs, the edge-intersection graphs of paths in a tree. This result generalizes a previously known characterization of equistable line graphs. Our approach is based on the combinatorial features of triangle graphs and general partition graphs. We also show that, in EPT graphs, testing whether a given clique is strong is co-NP-complete. We obtain this hardness result by first showing hardness of the problem of determining whether a given graph has a maximal matching disjoint from a given edge cut. As a positive result, we prove that the problem of testing whether a given clique is strong is polynomial in the class of local EPT graphs, which are defined as the edge intersection graphs of paths in a star and are known to coincide with the line graphs of multigraphs.Facultad de Ciencias ExactasConsejo Nacional de Investigaciones Científicas y Técnica
Strong cliques and equistability of EPT graphs
In this paper, we characterize the equistable graphs within the class of EPT graphs, the edge-intersection graphs of paths in a tree. This result generalizes a previously known characterization of equistable line graphs. Our approach is based on the combinatorial features of triangle graphs and general partition graphs. We also show that, in EPT graphs, testing whether a given clique is strong is co-NP-complete. We obtain this hardness result by first showing hardness of the problem of determining whether a given graph has a maximal matching disjoint from a given edge cut. As a positive result, we prove that the problem of testing whether a given clique is strong is polynomial in the class of local EPT graphs, which are defined as the edge intersection graphs of paths in a star and are known to coincide with the line graphs of multigraphs.Facultad de Ciencias ExactasConsejo Nacional de Investigaciones Científicas y Técnica