3 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 Minimum Dominating Sets in cubic and (claw,H)-free graphs
Given a graph , is a dominating set if every is adjacent to an element of . The Minimum Dominating Set
problem asks for a dominating set with minimum cardinality. It is well known
that its decision version is -complete even when is a claw-free graph.
We give a complexity dichotomy for the Minimum Dominating Set problem for the
class of -free graphs when has at most six vertices. In an
intermediate step we show that the Minimum Dominating Set problem is
-complete for cubic graphs
Hardness and approximation of minimum maximal matchings
In this paper, we consider the minimum maximal matching problem in some classes of graphs such as regular graphs. We show that the minimum maximal matching problem is NP-hard even in regular bipartite graphs, and a polynomial time exact algorithm is given for almost complete regular bipartite graphs. From the approximation point of view, it is well known that any maximal matching guarantees the approximation ratio of 2 but surprisingly very few improvements have been obtained. In this paper we give improved approximation ratios for several classes of graphs. For example any algorithm is shown to guarantee an approximation ratio of (2-o(1)) in graphs with high average degree. We also propose an algorithm guaranteeing for any graph of maximum degree Δ an approximation ratio of (2-1/Δ), which slightly improves the best known results. In addition, we analyse a natural linear-time greedy algorithm guaranteeing a ratio of (2-23/18k) in k-regular graphs admitting a perfect matching