7,097 research outputs found
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
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
Enumeration of maximum matchings of graphs
Counting maximum matchings in a graph is of great interest in statistical
mechanics,
solid-state chemistry, theoretical computer science, mathematics, among other
disciplines. However, it is a challengeable problem to explicitly determine the
number of maximum matchings of general graphs. In this paper, using
Gallai-Edmonds structure theorem, we derive a computing formula for the number
of maximum matching in a graph. According to the formula, we obtain an
algorithm to enumerate maximum matchings of a graph. In particular, The formula
implies that computing the number of maximum matchings of a graph is converted
to compute the number of perfect matchings of some induced subgraphs of the
graph. As an application, we calculate the number of maximum matchings of opt
trees. The result extends a conclusion obtained by Heuberger and Wagner[C.
Heuberger, S. Wagner, The number of maximum matchings in a tree, Discrete Math.
311 (2011) 2512--2542]
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
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