1,152 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
On Almost Well-Covered Graphs of Girth at Least 6
We consider a relaxation of the concept of well-covered graphs, which are
graphs with all maximal independent sets of the same size. The extent to which
a graph fails to be well-covered can be measured by its independence gap,
defined as the difference between the maximum and minimum sizes of a maximal
independent set in . While the well-covered graphs are exactly the graphs of
independence gap zero, we investigate in this paper graphs of independence gap
one, which we also call almost well-covered graphs. Previous works due to
Finbow et al. (1994) and Barbosa et al. (2013) have implications for the
structure of almost well-covered graphs of girth at least for . We focus on almost well-covered graphs of girth at least . We show
that every graph in this class has at most two vertices each of which is
adjacent to exactly leaves. We give efficiently testable characterizations
of almost well-covered graphs of girth at least having exactly one or
exactly two such vertices. Building on these results, we develop a
polynomial-time recognition algorithm of almost well-covered
-free graphs
Graphs of Edge-Intersecting Non-Splitting Paths in a Tree: Representations of Holes-Part II
Given a tree and a set P of non-trivial simple paths on it, VPT(P) is the VPT
graph (i.e. the vertex intersection graph) of the paths P, and EPT(P) is the
EPT graph (i.e. the edge intersection graph) of P. These graphs have been
extensively studied in the literature. Given two (edge) intersecting paths in a
graph, their split vertices is the set of vertices having degree at least 3 in
their union. A pair of (edge) intersecting paths is termed non-splitting if
they do not have split vertices (namely if their union is a path). We define
the graph ENPT(P) of edge intersecting non-splitting paths of a tree, termed
the ENPT graph, as the graph having a vertex for each path in P, and an edge
between every pair of vertices representing two paths that are both
edge-intersecting and non-splitting. A graph G is an ENPT graph if there is a
tree T and a set of paths P of T such that G=ENPT(P), and we say that is
a representation of G.
Our goal is to characterize the representation of chordless ENPT cycles
(holes). To achieve this goal, we first assume that the EPT graph induced by
the vertices of an ENPT hole is given. In [2] we introduce three assumptions
(P1), (P2), (P3) defined on EPT, ENPT pairs of graphs. In the same study, we
define two problems HamiltonianPairRec, P3-HamiltonianPairRec and characterize
the representations of ENPT holes that satisfy (P1), (P2), (P3).
In this work, we continue our work by relaxing these three assumptions one by
one. We characterize the representations of ENPT holes satisfying (P3) by
providing a polynomial-time algorithm to solve P3-HamiltonianPairRec. We also
show that there does not exist a polynomial-time algorithm to solve
HamiltonianPairRec, unless P=NP
- …