17 research outputs found
The travelling preacher, projection, and a lower bound for the stability number of a graph
AbstractThe coflow min–max equality is given a travelling preacher interpretation, and is applied to give a lower bound on the maximum size of a set of vertices, no two of which are joined by an edge
Sink-Stable Sets of Digraphs
We introduce the notion of sink-stable sets of a digraph and prove a min-max
formula for the maximum cardinality of the union of k sink-stable sets. The
results imply a recent min-max theorem of Abeledo and Atkinson on the Clar
number of bipartite plane graphs and a sharpening of Minty's coloring theorem.
We also exhibit a link to min-max results of Bessy and Thomasse and of Sebo on
cyclic stable sets
Sink-Stable Sets of Digraphs
We introduce the notion of sink-stable sets of a digraph and prove a min-max formula for the maximum cardinality of the union of k sink-stable sets. The results imply a recent min-max theorem of Abeledo and Atkinson on the Clar number of bipartite plane graphs and a sharpening of Minty’s coloring theorem. We also exhibit a link to min-max results of Bessy and Thomasse ́ and of Sebő on cyclic stable sets
A network flow approach to a common generalization of Clar and Fries numbers
Clar number and Fries number are two thoroughly investigated parameters of
plane graphs emerging from mathematical chemistry to measure stability of
organic molecules. We consider first a common generalization of these two
concepts for bipartite plane graphs, and then extend it to a framework on
general (not necessarily planar) directed graphs. The corresponding
optimization problem can be transformed into a maximum weight feasible tension
problem which is the linear programming dual of a minimum cost network flow (or
circulation) problem. Therefore the approach gives rise to a min-max theorem
and to a strongly polynomial algorithm that relies exclusively on standard
network flow subroutines. In particular, we give the first network flow based
algorithm for an optimal Fries structure and its variants
Basic Neutrosophic Algebraic Structures and their Application to Fuzzy and Neutrosophic Models
The involvement of uncertainty of varying degrees when the total of the
membership degree exceeds one or less than one, then the newer mathematical
paradigm shift, Fuzzy Theory proves appropriate. For the past two or more
decades, Fuzzy Theory has become the potent tool to study and analyze
uncertainty involved in all problems. But, many real-world problems also abound
with the concept of indeterminacy. In this book, the new, powerful tool of
neutrosophy that deals with indeterminacy is utilized. Innovative neutrosophic
models are described. The theory of neutrosophic graphs is introduced and
applied to fuzzy and neutrosophic models. This book is organized into four
chapters. In Chapter One we introduce some of the basic neutrosophic algebraic
structures essential for the further development of the other chapters. Chapter
Two recalls basic graph theory definitions and results which has interested us
and for which we give the neutrosophic analogues. In this chapter we give the
application of graphs in fuzzy models. An entire section is devoted for this
purpose. Chapter Three introduces many new neutrosophic concepts in graphs and
applies it to the case of neutrosophic cognitive maps and neutrosophic
relational maps. The last section of this chapter clearly illustrates how the
neutrosophic graphs are utilized in the neutrosophic models. The final chapter
gives some problems about neutrosophic graphs which will make one understand
this new subject.Comment: 149 pages, 130 figure
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum