543 research outputs found
Maximal induced matchings in triangle-free graphs
An induced matching in a graph is a set of edges whose endpoints induce a
-regular subgraph. It is known that any -vertex graph has at most
maximal induced matchings, and this bound is best
possible. We prove that any -vertex triangle-free graph has at most maximal induced matchings, and this bound is attained by any
disjoint union of copies of the complete bipartite graph . Our result
implies that all maximal induced matchings in an -vertex triangle-free graph
can be listed in time , yielding the fastest known algorithm for
finding a maximum induced matching in a triangle-free graph.Comment: 17 page
Between proper and strong edge-colorings of subcubic graphs
In a proper edge-coloring the edges of every color form a matching. A
matching is induced if the end-vertices of its edges induce a matching. A
strong edge-coloring is an edge-coloring in which the edges of every color form
an induced matching. We consider intermediate types of edge-colorings, where
edges of some colors are allowed to form matchings, and the remaining form
induced matchings. Our research is motivated by the conjecture proposed in a
recent paper of Gastineau and Togni on S-packing edge-colorings (On S-packing
edge-colorings of cubic graphs, Discrete Appl. Math. 259 (2019), 63-75)
asserting that by allowing three additional induced matchings, one is able to
save one matching color. We prove that every graph with maximum degree 3 can be
decomposed into one matching and at most 8 induced matchings, and two matchings
and at most 5 induced matchings. We also show that if a graph is in class I,
the number of induced matchings can be decreased by one, hence confirming the
above-mentioned conjecture for class I graphs
Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs
A bipartite graph is convex if the vertices in can be
linearly ordered such that for each vertex , the neighbors of are
consecutive in the ordering of . An induced matching of is a
matching such that no edge of connects endpoints of two different edges of
. We show that in a convex bipartite graph with vertices and
weighted edges, an induced matching of maximum total weight can be computed in
time. An unweighted convex bipartite graph has a representation of
size that records for each vertex the first and last neighbor
in the ordering of . Given such a compact representation, we compute an
induced matching of maximum cardinality in time.
In convex bipartite graphs, maximum-cardinality induced matchings are dual to
minimum chain covers. A chain cover is a covering of the edge set by chain
subgraphs, that is, subgraphs that do not contain induced matchings of more
than one edge. Given a compact representation, we compute a representation of a
minimum chain cover in time. If no compact representation is given, the
cover can be computed in time.
All of our algorithms achieve optimal running time for the respective problem
and model. Previous algorithms considered only the unweighted case, and the
best algorithm for computing a maximum-cardinality induced matching or a
minimum chain cover in a convex bipartite graph had a running time of
Efficient edge domination in regular graphs
An induced matching of a graph G is a matching having no two edges joined by an edge. An efficient edge dominating set
of G is an induced matching M such that every other edge of G is adjacent to some edge in M. We relate maximum induced
matchings and efficient edge dominating sets, showing that efficient edge dominating sets are maximum induced matchings, and
that maximum induced matchings on regular graphs with efficient edge dominating sets are efficient edge dominating sets. A
necessary condition for the existence of efficient edge dominating sets in terms of spectra of graphs is established. We also prove
that, for arbitrary fixed p 3, deciding on the existence of efficient edge dominating sets on p-regular graphs is NP-complet
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