13 research outputs found
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
Robustness: a New Form of Heredity Motivated by Dynamic Networks
We investigate a special case of hereditary property in graphs, referred to
as {\em robustness}. A property (or structure) is called robust in a graph
if it is inherited by all the connected spanning subgraphs of . We motivate
this definition using two different settings of dynamic networks. The first
corresponds to networks of low dynamicity, where some links may be permanently
removed so long as the network remains connected. The second corresponds to
highly-dynamic networks, where communication links appear and disappear
arbitrarily often, subject only to the requirement that the entities are
temporally connected in a recurrent fashion ({\it i.e.} they can always reach
each other through temporal paths). Each context induces a different
interpretation of the notion of robustness.
We start by motivating the definition and discussing the two interpretations,
after what we consider the notion independently from its interpretation, taking
as our focus the robustness of {\em maximal independent sets} (MIS). A graph
may or may not admit a robust MIS. We characterize the set of graphs \forallMIS
in which {\em all} MISs are robust. Then, we turn our attention to the graphs
that {\em admit} a robust MIS (\existsMIS). This class has a more complex
structure; we give a partial characterization in terms of elementary graph
properties, then a complete characterization by means of a (polynomial time)
decision algorithm that accepts if and only if a robust MIS exists. This
algorithm can be adapted to construct such a solution if one exists
Some problems related to keys and the Boyce-Codd normal form
The aim of this paper is to investigate the connections between minimal keys and antikeys for special Sperner-systems by hypergraphs. The Boyce-Codd normal form and some related problems are also studied in this paper
Several identities for the characteristic polynomial of a combinatorial geometry
In this paper we explore a research problem of Greene: to find inequalities for the Möbius function which become equalities in the presence of modularity. We replace these inequalities with identities and give combinatorial interpretations for the difference
Cyclic and constacyclic codes over a non-chain ring
oai:ojs2.jacodesmath.com:article/1In this study, we consider linear and especially cyclic codes over the non-chain ring where is a prime. This is a generalization of the case Further, in this work the structure of constacyclic codes are studied as well. This study takes advantage mainly from a Gray map which preserves the distance between codes over this ring and -ary codes and moreover this map enlightens the structure of these codes. Furthermore, a MacWilliams type identity is presented together with some illustrative examples
Recommended from our members
A Logical and Computational Methodology for Exploring Systems of Phonotactic Constraints
We introduce a methodology built around a logical analysis component based on a hierarchy of classes of Subregular constraints characterized by the kinds of features of a string a mechanism must be sensitive to in order to determine whether it satisfies the constraint, and a computational component built around a publicly-available interactive workbench that implements, based on the equivalence between logical formulae and finite-state automata, a theorem prover for these logics (even algorithmically extracting certain classes of constraints), wherein the alternation between these logical and computational analyses can provide useful insight more easily than using either in isolation