10 research outputs found
A Set and Collection Lemma
A set S is independent if no two vertices from S are adjacent. In this paper
we prove that if F is a collection of maximum independent sets of a graph, then
there is a matching from S-{intersection of all members of F} into {union of
all members of F}-S, for every independent set S. Based on this finding we give
alternative proofs for a number of well-known lemmata, as the "Maximum Stable
Set Lemma" due to Claude Berge and the "Clique Collection Lemma" due to
Andr\'as Hajnal.Comment: 7 pages, 3 figure
On Konig-Egervary Collections of Maximum Critical Independent Sets
Let G be a simple graph with vertex set V(G). A set S is independent if no
two vertices from S are adjacent. The graph G is known to be a Konig-Egervary
if alpha(G)+mu(G)= |V(G)|, where alpha(G) denotes the size of a maximum
independent set and mu(G) is the cardinality of a maximum matching. The number
d(X)= |X|-|N(X)| is the difference of X, and an independent set A is critical
if d(A) = max{d(I):I is an independent set in G} (Zhang; 1990). Let Omega(G)
denote the family of all maximum independent sets. Let us say that a family
Gamma of independent sets is a Konig-Egervary collection if |Union of Gamma| +
|Intersection of Gamma| = 2alpha(G) (Jarden, Levit, Mandrescu; 2015). In this
paper, we show that if the family of all maximum critical independent sets of a
graph G is a Konig-Egervary collection, then G is a Konig-Egervary graph. It
generalizes one of our conjectures recently validated in (Short; 2015).Comment: 9 pages, 3 figure
On some conjectures concerning critical independent sets of a graph
Let be a simple graph with vertex set . A set is
independent if no two vertices from are adjacent. For ,
the difference of is and an independent set is
critical if (possibly ). Let and
be the intersection and union, respectively, of all maximum
size critical independent sets in . In this paper, we will give two new
characterizations of K\"{o}nig-Egerv\'{a}ry graphs involving
and . We also prove a related lower bound
for the independence number of a graph. This work answers several conjectures
posed by Jarden, Levit, and Mandrescu.Comment: 10 pages, 3 figure
Critical and Maximum Independent Sets of a Graph
Let G be a simple graph with vertex set V(G). A subset S of V(G) is
independent if no two vertices from S are adjacent. By Ind(G) we mean the
family of all independent sets of G while core(G) and corona(G) denote the
intersection and the union of all maximum independent sets, respectively. The
number d(X)= |X|-|N(X)| is the difference of the set of vertices X, and an
independent set A is critical if d(A)=max{d(I):I belongs to Ind(G)} (Zhang,
1990). Let ker(G) and diadem(G) be the intersection and union, respectively, of
all critical independent sets of G (Levit and Mandrescu, 2012). In this paper,
we present various connections between critical unions and intersections of
maximum independent sets of a graph. These relations give birth to new
characterizations of Koenig-Egervary graphs, some of them involving ker(G),
core(G), corona(G), and diadem(G).Comment: 12 pages, 9 figures. arXiv admin note: substantial text overlap with
arXiv:1407.736
On the Critical Difference of Almost Bipartite Graphs
A set is \textit{independent} in a graph if no two vertices from are adjacent. The \textit{independence number}
is the cardinality of a maximum independent set, while is
the size of a maximum matching in . If equals the order
of , then is called a \textit{K\"{o}nig-Egerv\'{a}ry graph
}\cite{dem,ster}. The number is called the
\textit{critical difference} of \cite{Zhang} (where ). It is known
that holds for every graph
\cite{Levman2011a,Lorentzen1966,Schrijver2003}. In \cite{LevMan5} it was shown
that is true for every K\"{o}nig-Egerv\'{a}ry graph.
A graph is \textit{(i)} \textit{unicyclic} if it has a unique cycle,
\textit{(ii)} \textit{almost bipartite} if it has only one odd cycle. It was
conjectured in \cite{LevMan2012a,LevMan2013a} and validated in
\cite{Bhattacharya2018} that holds for every unicyclic
non-K\"{o}nig-Egerv\'{a}ry graph .
In this paper we prove that if is an almost bipartite graph of order
, then . Moreover, for each of these two values, we
characterize the corresponding graphs. Further, using these findings, we show
that the critical difference of an almost bipartite graph satisfies where by \textrm{core} we
mean the intersection of all maximum independent sets.Comment: 12 pages, 5 figures. arXiv admin note: text overlap with
arXiv:1102.472
Critical Independent Sets of a Graph
Let be a simple graph with vertex set . A set
is independent if no two vertices from are
adjacent, and by we mean the family of all independent sets
of .
The number is the difference of , and a
set is critical if (Zhang, 1990).
Let us recall the following definitions:
= {S : S is a maximum independent
set}.
= {S :S is a maximum independent
set}.
= {S : S is a critical independent set}.
= {S : S is a critical independent set}.
In this paper we present various structural properties of ,
in relation with , , and .Comment: 15 pages; 12 figures. arXiv admin note: substantial text overlap with
arXiv:1102.113
Monotonic Properties of Collections of Maximum Independent Sets of a Graph
Let G be a simple graph with vertex set V(G). A subset S of V(G) is
independent if no two vertices from S are adjacent. The graph G is known to be
a Konig-Egervary if alpha(G) + mu(G)= |V(G)|, where alpha(G) denotes the size
of a maximum independent set and mu(G) is the cardinality of a maximum
matching. Let Omega(G) denote the family of all maximum independent sets, and f
be the function from the set of subcollections Gamma of Omega(G) such that
f(Gamma) = (the cardinality of the union of elements of Gamma) + (the
cardinality of the intersection of elements of Gamma). Our main finding claims
that f is "<<"-increasing, where the preorder {Gamma1} << {Gamma2} means that
the union of all elements of {Gamma1} is a subset of the union of all elements
of {Gamma2}, while the intersection of all elements of {Gamma2} is a subset of
the intersection of all elements of {Gamma1}. Let us say that a family {Gamma}
is a Konig-Egervary collection if f(Gamma) = 2*alpha(G). We conclude with the
observation that for every graph G each subcollection of a Konig-Egervary
collection is Konig-Egervary as well.Comment: 15 pages, 7 figure
On an Annihilation Number Conjecture
Let denote the cardinality of a maximum independent set, while
be the size of a maximum matching in the graph .
If , then is a
K\"onig-Egerv\'ary graph. If is the
degree sequence of , then the annihilation number of
is the largest integer such that (Pepper 2004, Pepper 2009). A set satisfying
is an annihilation
set, if, in addition, , for every vertex , then is a
maximal annihilation set in .
In (Larson & Pepper 2011) it was conjectured that the following assertions
are equivalent:
(i) ;
(ii) is a K\"onig-Egerv\'ary graph and every maximum independent set is a
maximal annihilating set.
In this paper, we prove that the implication "(i) (ii)" is
correct, while for the opposite direction we provide a series of generic
counterexamples.
Keywords: maximum independent set, matching, tree, bipartite graph,
K\"onig-Egerv\'ary graph, annihilation set, annihilation number.Comment: 17 pages, 11 figure
Critical Sets in Bipartite Graphs
Let G=(V,E) be a graph. A set S is independent if no two vertices from S are
adjacent, alpha(G) is the size of a maximum independent set, and core(G) is the
intersection of all maximum independent sets. The number d(X)=|X|-|N(X)| is the
difference of the set X, and d_{c}(G)=max{d(I):I is an independent set} is
called the critical difference of G. A set X is critical if d(X)=d_{c}(G). For
a graph G we define ker(G) as the intersection of all critical independent
sets, while diadem(G) is the union of all critical independent sets. For a
bipartite graph G=(A,B,E), with bipartition {A,B}, Ore defined delta(X)=d(X)
for every subset X of A, while delta_0(A)=max{delta(X):X is a subset of A}.
Similarly is defined delta_0(B). In this paper we prove that for every
bipartite graph G=(A,B,E) the following assertions hold:
d_{c}(G)=delta_0(A)+delta_0(B); ker(G)=core(G);
|ker(G)|+|diadem(G)|=2*alpha(G).Comment: 13 pages, 8 figure
Critical sets, crowns, and local maximum independent sets
A set is independent (or stable) if no two vertices from
are adjacent, and by we mean the set of all independent
sets of .
A set is critical (and we write ) if
, where
denotes the neighborhood of .
If and there is a matching from into , then
is a crown, and we write .
Let be the family of all local maximum independent sets of graph
, i.e., if is a maximum independent set in the subgraph
induced by .
In this paper we show that
are true for every graph. In addition, we present some
classes of graphs where these families coincide and form greedoids or even more
general set systems that we call augmentoids.Comment: 19 pages, 11 figure