7 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
Weighted Well-Covered Claw-Free Graphs
A graph G is well-covered if all its maximal independent sets are of the same
cardinality. Assume that a weight function w is defined on its vertices. Then G
is w-well-covered if all maximal independent sets are of the same weight. For
every graph G, the set of weight functions w such that G is w-well-covered is a
vector space. Given an input claw-free graph G, we present an O(n^6)algortihm,
whose input is a claw-free graph G, and output is the vector space of weight
functions w, for which G is w-well-covered. A graph G is equimatchable if all
its maximal matchings are of the same cardinality. Assume that a weight
function w is defined on the edges of G. Then G is w-equimatchable if all its
maximal matchings are of the same weight. For every graph G, the set of weight
functions w such that G is w-equimatchable is a vector space. We present an
O(m*n^4 + n^5*log(n)) algorithm which receives an input graph G, and outputs
the vector space of weight functions w such that G is w-equimatchable.Comment: 14 pages, 1 figur
Some Extremal And Structural Problems In Graph Theory
This work considers three main topics. In Chapter 2, we deal with König-Egerváry graphs. We will give two new characterizations of König-Egerváry graphs as well as prove a related lower bound for the independence number of a graph. In Chapter 3, we study joint degree vectors (JDV). A problem arising from statistics is to determine the maximum number of non-zero elements of a JDV. We provide reasonable lower and upper bounds for this maximum number. Lastly, in Chapter 4 we study a problem in chemical graph theory. In particular, we characterize extremal cases for the number of maximal matchings in two linear polymers of chemical interest: the polyspiro chains and benzenoid chains. We also enumerate maximal matchings in several classes of these linear polymers and use the obtained results to determine the asymptotic behavior of these matchings