646 research outputs found
The number of independent sets in unicyclic graphs
AbstractIn this paper, we determine upper and lower bounds for the number of independent sets in a unicyclic graph in terms of its order. This gives an upper bound for the number of independent sets in a connected graph which contains at least one cycle. We also determine the upper bound for the number of independent sets in a unicyclic graph in terms of order and girth. In each case, we characterize the extremal graphs
On the largest real root of the independence polynomial of a unicyclic graph
The independence polynomial of a graph , denoted , is the
generating polynomial for the number of independent sets of each size. The
roots of are called the \textit{independence roots} of . It is
known that for every graph , the independence root of smallest modulus,
denoted , is real. The relation on the set of all graphs is
defined as follows, if and only if
We find the maximum and minimum connected unicyclic and connected
well-covered unicyclic graphs of a given order with respect to . This
extends recent work by Oboudi where the maximum and minimum trees of a given
order were determined and also answers an open question posed in the same work.
Corollaries of our results give the graphs that minimize and maximize
among all connected (well-covered) unicyclic graphs. We also answer more open
questions posed by Oboudi and disprove a related conjecture due to Levit and
Mandrescu
On the Core of a Unicyclic Graph
A set S is independent in a graph G if no two vertices from S are adjacent.
By core(G) we mean the intersection of all maximum independent sets. The
independence number alpha(G) is the cardinality of a maximum independent set,
while mu(G) is the size of a maximum matching in G. A connected graph having
only one cycle, say C, is a unicyclic graph. In this paper we prove that if G
is a unicyclic graph of order n and n-1 = alpha(G) + mu(G), then core(G)
coincides with the union of cores of all trees in G-C.Comment: 8 pages, 5 figure
Computing Unique Maximum Matchings in O(m) time for Konig-Egervary Graphs and Unicyclic Graphs
Let alpha(G) denote the maximum size of an independent set of vertices and
mu(G) be the cardinality of a maximum matching in a graph G. A matching
saturating all the vertices is perfect. If alpha(G) + mu(G) equals the number
of vertices of G, then it is called a Konig-Egervary graph. A graph is
unicyclic if it has a unique cycle.
In 2010, Bartha conjectured that a unique perfect matching, if it exists, can
be found in O(m) time, where m is the number of edges.
In this paper we validate this conjecture for Konig-Egervary graphs and
unicylic graphs. We propose a variation of Karp-Sipser leaf-removal algorithm
(Karp and Spiser, 1981), which ends with an empty graph if and only if the
original graph is a Konig-Egervary graph with a unique perfect matching
obtained as an output as well.
We also show that a unicyclic non-bipartite graph G may have at most one
perfect matching, and this is the case where G is a Konig-Egervary graph.Comment: 10 pages, 5 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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