43,470 research outputs found
Edge-Stable Equimatchable Graphs
A graph is \emph{equimatchable} if every maximal matching of has the
same cardinality. We are interested in equimatchable graphs such that the
removal of any edge from the graph preserves the equimatchability. We call an
equimatchable graph \emph{edge-stable} if , that is the
graph obtained by the removal of edge from , is also equimatchable for
any . After noticing that edge-stable equimatchable graphs are
either 2-connected factor-critical or bipartite, we characterize edge-stable
equimatchable graphs. This characterization yields an time recognition algorithm. Lastly, we introduce and shortly
discuss the related notions of edge-critical, vertex-stable and vertex-critical
equimatchable graphs. In particular, we emphasize the links between our work
and the well-studied notion of shedding vertices, and point out some open
questions
Nonbipartite Dulmage-Mendelsohn Decomposition for Berge Duality
The Dulmage-Mendelsohn decomposition is a classical canonical decomposition
in matching theory applicable for bipartite graphs, and is famous not only for
its application in the field of matrix computation, but also for providing a
prototypal structure in matroidal optimization theory. The Dulmage-Mendelsohn
decomposition is stated and proved using the two color classes, and therefore
generalizing this decomposition for nonbipartite graphs has been a difficult
task. In this paper, we obtain a new canonical decomposition that is a
generalization of the Dulmage-Mendelsohn decomposition for arbitrary graphs,
using a recently introduced tool in matching theory, the basilica
decomposition. Our result enables us to understand all known canonical
decompositions in a unified way. Furthermore, we apply our result to derive a
new theorem regarding barriers. The duality theorem for the maximum matching
problem is the celebrated Berge formula, in which dual optimizers are known as
barriers. Several results regarding maximal barriers have been derived by known
canonical decompositions, however no characterization has been known for
general graphs. In this paper, we provide a characterization of the family of
maximal barriers in general graphs, in which the known results are developed
and unified
Metric characterization of cluster dynamics on the Sierpinski gasket
We develop and implement an algorithm for the quantitative characterization
of cluster dynamics occurring on cellular automata defined on an arbitrary
structure. As a prototype for such systems we focus on the Ising model on a
finite Sierpsinski Gasket, which is known to possess a complex thermodynamic
behavior. Our algorithm requires the projection of evolving configurations into
an appropriate partition space, where an information-based metrics (Rohlin
distance) can be naturally defined and worked out in order to detect the
changing and the stable components of clusters. The analysis highlights the
existence of different temperature regimes according to the size and the rate
of change of clusters. Such regimes are, in turn, related to the correlation
length and the emerging "critical" fluctuations, in agreement with previous
thermodynamic analysis, hence providing a non-trivial geometric description of
the peculiar critical-like behavior exhibited by the system. Moreover, at high
temperatures, we highlight the existence of different time scales controlling
the evolution towards chaos.Comment: 20 pages, 8 figure
Problems on Matchings and Independent Sets of a Graph
Let be a finite simple graph. For , the difference of
, where is the neighborhood of and is called the critical difference of . is
called a critical set if equals the critical difference and ker is
the intersection of all critical sets. It is known that ker is an
independent (vertex) set of . diadem is the union of all critical
independent sets. An independent set is an inclusion minimal set with if no proper subset of has positive difference.
A graph is called K\"onig-Egerv\'ary if the sum of its independence
number () and matching number () equals . It is
known that bipartite graphs are K\"onig-Egerv\'ary.
In this paper, we study independent sets with positive difference for which
every proper subset has a smaller difference and prove a result conjectured by
Levit and Mandrescu in 2013. The conjecture states that for any graph, the
number of inclusion minimal sets with is at least the critical
difference of the graph. We also give a short proof of the inequality
kerdiadem (proved by Short in 2016).
A characterization of unicyclic non-K\"onig-Egerv\'ary graphs is also
presented and a conjecture which states that for such a graph , the critical
difference equals , is proved.
We also make an observation about ker using Edmonds-Gallai Structure
Theorem as a concluding remark.Comment: 18 pages, 2 figure
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