138 research outputs found
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
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
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