15 research outputs found
On the approximability of the maximum induced matching problem
In this paper we consider the approximability of the maximum induced matching problem (MIM). We give an approximation algorithm with asymptotic performance ratio <i>d</i>-1 for MIM in <i>d</i>-regular graphs, for each <i>d</i>≥3. We also prove that MIM is APX-complete in <i>d</i>-regular graphs, for each <i>d</i>≥3
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
Edge open packing: complexity, algorithmic aspects, and bounds
Given a graph , two edges are said to have a common
edge if joins an endvertex of to an endvertex of . A
subset is an edge open packing set in if no two edges of
have a common edge in , and the maximum cardinality of such a set in
is called the edge open packing number, , of . In this
paper, we prove that the decision version of the edge open packing number is
NP-complete even when restricted to graphs with universal vertices, Eulerian
bipartite graphs, and planar graphs with maximum degree , respectively. In
contrast, we present a linear-time algorithm that computes the edge open
packing number of a tree. We also resolve two problems posed in the seminal
paper [Edge open packing sets in graphs, RAIRO-Oper.\ Res.\ 56 (2022)
3765--3776]. Notably, we characterize the graphs that attain the upper
bound , and provide lower and upper bounds for
the edge-deleted subgraph of a graph and establish the corresponding
realization result.Comment: 12 pages, 1 figur
Parameterized Complexity of Perfectly Matched Sets
For an undirected graph G, a pair of vertex disjoint subsets (A, B) is a pair of perfectly matched sets if each vertex in A (resp. B) has exactly one neighbor in B (resp. A). In the above, the size of the pair is |A| (= |B|). Given a graph G and a positive integer k, the Perfectly Matched Sets problem asks whether there exists a pair of perfectly matched sets of size at least k in G. This problem is known to be NP-hard on planar graphs and W[1]-hard on general graphs, when parameterized by k. However, little is known about the parameterized complexity of the problem in restricted graph classes. In this work, we study the problem parameterized by k, and design FPT algorithms for: i) apex-minor-free graphs running in time 2^O(?k)? n^O(1), and ii) K_{b,b}-free graphs. We obtain a linear kernel for planar graphs and k^?(d)-sized kernel for d-degenerate graphs. It is known that the problem is W[1]-hard on chordal graphs, in fact on split graphs, parameterized by k. We complement this hardness result by designing a polynomial-time algorithm for interval graphs
Induced Matching below Guarantees: Average Paves the Way for Fixed-Parameter Tractability
In this work, we study the Induced Matching problem: Given an undirected
graph and an integer , is there an induced matching of size at
least ? An edge subset is an induced matching in if is a
matching such that there is no edge between two distinct edges of . Our work
looks into the parameterized complexity of Induced Matching with respect to
"below guarantee" parameterizations. We consider the parameterization for an upper bound on the size of any induced matching. For instance,
any induced matching is of size at most where is the number of
vertices, which gives us a parameter . In fact, there is a
straightforward -time algorithm for Induced
Matching [Moser and Thilikos, J. Discrete Algorithms]. Motivated by this, we
ask: Is Induced Matching FPT for a parameter smaller than ? In
search for such parameters, we consider and ,
where is the maximum matching size and is the maximum
independent set size of . We find that Induced Matching is presumably not
FPT when parameterized by or . In contrast to
these intractability results, we find that taking the average of the two helps
-- our main result is a branching algorithm that solves Induced Matching in
time. Our algorithm makes use
of the Gallai-Edmonds decomposition to find a structure to branch on