Induced Matching Research Articles

Between proper and strong edge‐colorings of subcubic graphs

AbstractIn 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 ‐packing edge‐colorings asserting that, for subcubic graphs, 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 eight induced matchings, and two matchings and at most five 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.

Latest Algorithms on Particular Graph Classes

Many optimization problems such as Maximum Independent Set, Maximum Clique, Minimum Clique Cover and Maximum Induced Matching are NP-hard on general graphs. However, they could be solved in polynomial time when restricted to some particular graph classes such as comparability and co-comparability graph classes. In this paper, we summarize the latest algorithms solving some classical NP-hard problems on some graph classes over the years. Moreover, we apply the -redundant technique to obtain linear time O(j j) algorithms which find a Maximum Induced Matching on interval and circular-arc graphs. Inspired of these results, we have proposed some competitive programming problems for some programming contests in Vietnam in recent years.

Colorings with only rainbow arithmetic progressions

If we want to color $$1,2,\ldots ,n$$ with the property that all 3-term arithmetic progressions are rainbow (that is, their elements receive 3 distinct colors), then, obviously, we need to use at least n/2 colors. Surprisingly, much fewer colors suffice if we are allowed to leave a negligible proportion of integers uncolored. Specifically, we prove that there exist $$\alpha ,\beta <1$$ such that for every n, there is a subset A of $$\{1,2,\ldots ,n\}$$ of size at least $$n-n^{\alpha }$$ , the elements of which can be colored with $$n^{\beta }$$ colors with the property that every 3-term arithmetic progression in A is rainbow. Moreover, $$\beta $$ can be chosen to be arbitrarily small. Our result can be easily extended to k-term arithmetic progressions for any $$k\ge 3$$ . As a corollary, we obtain a simple proof of the following result of Alon, Moitra, and Sudakov, which can be used to design efficient communication protocols over shared directional multi-channels. There exist $$\alpha ',\beta '<2$$ such that for every n, there is a graph with n vertices and at least $$\left( {\begin{array}{c}n 2\end{array}}\right) -n^{\alpha '}$$ edges, whose edge set can be partitioned into at most $$n^{\beta '}$$ induced matchings.

Maximum weight induced matching in some subclasses of bipartite graphs

A subset $$M\subseteq E$$ of edges of a graph $$G=(V,E)$$ is called a matching in G if no two edges in M share a common vertex. A matching M in G is called an induced matching if G[M], the subgraph of G induced by M, is the same as G[S], the subgraph of G induced by $$S=\{v \in V |$$ v is incident on an edge of $$M\}$$ . The Maximum Induced Matching problem is to find an induced matching of maximum cardinality. Given a graph G and a positive integer k, the Induced Matching Decision problem is to decide whether G has an induced matching of cardinality at least k. The Maximum Weight Induced Matching problem in a weighted graph $$G=(V,E)$$ in which the weight of each edge is a positive real number, is to find an induced matching such that the sum of the weights of its edges is maximum. It is known that the Induced Matching Decision problem and hence the Maximum Weight Induced Matching problem is known to be NP-complete for general graphs and bipartite graphs. In this paper, we strengthened this result by showing that the Induced Matching Decision problem is NP-complete for star-convex bipartite graphs, comb-convex bipartite graphs, and perfect elimination bipartite graphs, the subclasses of the class of bipartite graphs. On the positive side, we propose polynomial time algorithms for the Maximum Weight Induced Matching problem for circular-convex bipartite graphs and triad-convex bipartite graphs by making polynomial time reductions from the Maximum Weight Induced Matching problem in these graph classes to the Maximum Weight Induced Matching problem in convex bipartite graphs.

Maximum Induced Matching Algorithms via Vertex Ordering Characterizations

We study the maximum induced matching problem on a graph G. Induced matchings correspond to independent sets in $$L^2(G)$$, the square of the line graph of G. The problem is NP-complete on bipartite graphs. In this work, we show that for a number of graph families characterized by vertex orderings, almost all forbidden patterns on three vertices are preserved when taking the square of the line graph. These orderings can be computed in linear time in the size of the input graph. In particular, given $$\mathcal {G}$$ a graph class, and a graph $$G=(V,E) \in \mathcal {G}$$ with a corresponding vertex ordering $$\sigma $$ of V, one can produce (in linear time in the size of G) an ordering on the vertices of $$L^2(G)$$, that shows that $$L^2(G) \in \mathcal {G}$$ without computing the line graph or the square of the line graph of G. These results generalize and unify previous ones on showing closure under $$L^2(\cdot )$$ for various graph families. Furthermore, these orderings on $$L^2(G)$$ can be exploited algorithmically to compute a maximum induced matching on G faster. We illustrate this latter fact in the second half of the paper where we focus on cocomparability graphs, a large graph class that includes interval, permutation, trapezoid graphs, and co-graphs, and we present the first $$\mathcal {O}(mn)$$ time algorithm to compute a maximum weighted induced matching on cocomparability graphs; an improvement from the best known $$\mathcal {O}(n^4)$$ time algorithm for the unweighted case.

On Maximum Induced Matching Numbers of Special Grids

A subset $M$ of the edge set of a graph $G$ is an induced matching of $G$ if given any two $e_1,e_2 \in M$, none of the vertices on $e_1$ is adjacent to any of the vertices on $e_2$. Suppose that $MIM_G$, a positive integer, is the largest possible size of $M$ in $G$, then, $M$ is the maximum induced matching, $MIM$, of $G$ and $MIM_G$ is the maximum induced matching number of $G$. We obtain some upper bounds for the maximum induced matching numbers of some specific grids.

Integer Programming Formulations and Benders Decomposition for the Maximum Induced Matching Problem

We investigate the maximum induced matching problem (MIM), which is the problem of finding an induced matching having the largest cardinality on an undirected graph. The problem is known to be NP-hard for general graphs. We first propose a vertex-based integer programming formulation for MIM, which is more compact compared to an edge-based formulation found in the literature. We also introduce the maximum weight induced matching problem (MWIM), which generalizes MIM so that vertices and edges have weights. We adapt the edge-based formulation to MWIM, and propose a quadratic programming formulation of MWIM based on our vertex-based formulation. We then linearize our quadratic programming formulation, and devise a Benders decomposition algorithm that exploits a special structure of the linearized formulation. We also propose valid inequalities and formulation tightening procedures to improve the efficiency of our approach. Our computational tests on a large suite of randomly generated graphs show that our vertex-based formulation and decomposition approach significantly improve the solvability of MIM and MWIM, especially on dense graphs. The online appendix and data are available at https://doi.org/10.1287/ijoc.2017.0764 .

A Faster Algorithm for Maximum Induced Matchings on Circle Graphs

Circle graphs have applications to RNA bioinformatics, computational chemistry, and VLSI design. Additionally, many problems that are intractable on general graphs are efficient for circle graphs. This has driven research into algorithms for circle graphs. One well known graph problem is to find a maximum induced matching. This is NP-Hard, even for bipartite graphs. No algorithm for this problem that works directly on circle graphs has been proposed. However, since circle graphs are included in interval filament graphs, algorithms for this class can be applied to circle graphs. Unfortunately, this entails a large computational cost of $O(|V|^6)$ time. We propose an algorithm that operates directly on circle graphs, and requires only $O(|V|^3)$ time.

Exact algorithms for Maximum Induced Matching

This paper studies exact algorithms for the Maximum Induced Matching problem, in which an n-vertex graph is given and we are asked to find a set of maximum number of edges in the graph such that no pair of edges in the set have a common endpoint or are adjacent by another edge. This problem has applications in many different areas. We give several structural properties of the problem and show that the problem can be solved in O⁎(1.4231n) time and polynomial space or O⁎(1.3752n) time and exponential space.

On Minimum Maximal Distance-k Matchings

We study the computational complexity of several problems connected with the problems of finding a maximal distance-k matching of minimum cardinality or minimum weight. We introduce the class of k-equimatchable graphs which is an edge analogue of k-equipackable graphs. We prove that the recognition of k-equimatchable graphs is co-NP-complete for any fixed k≥2. We also prove that the problem of finding a minimum weight maximal distance-2l matching in chordal graphs is hard to approximate within a factor of εln|V(G)| for a fixed ε unless P=NP. Finally, we show NP-hardness of the minimum maximal induced matching problem in several restricted graph classes.

Graphs with maximal induced matchings of the same size

A graph is well-indumatched if all its maximal induced matchings are of the same size. We first prove that recognizing whether a graph is well-indumatched is a co-NP-complete problem even for (2P5,K1,5)-free graphs. We then show that decision problems Independent Dominating Set, Independent Set, and Dominating Set are NP-complete for the class of well-indumatched graphs. We also show that this class is a co-indumatching hereditary class, i.e., it is closed under deleting the end-vertices of an induced matching along with their neighborhoods, and we characterize well-indumatched graphs in terms of forbidden co-indumatching subgraphs. We prove that recognizing a co-indumatching subgraph is an NP-complete problem. We introduce a perfectly well-indumatched graph, in which every induced subgraph is well-indumatched, and characterize the class of these graphs in terms of forbidden induced subgraphs. Finally, we show that the weighted versions of problems Independent Dominating Set and Independent Set can be solved in polynomial time for perfectly well-indumatched graphs, but problem Dominating Set is NP-complete.

Exact Algorithms for Minimum Weighted Dominating Induced Matching

Say that an edge of a graph G dominates itself and every other edge sharing a vertex of it. An edge dominating set of a graph $$G=(V,E)$$G=(V,E) is a subset of edges $$E' \subseteq E$$Eź⊆E which dominates all edges of G. In particular, if every edge of G is dominated by exactly one edge of $$E'$$Eź then $$E'$$Eź is a dominating induced matching. It is known that not every graph admits a dominating induced matching, while the problem to decide if it does admit it is NP-complete. In this paper we consider the problems of counting the number of dominating induced matchings and finding a minimum weighted dominating induced matching, if any, of a graph with weighted edges. We describe three exact algorithms for general graphs. The first runs in linear time for a given vertex dominating set of fixed size of the graph. The second runs in polynomial time if the graph admits a polynomial number of maximal independent sets. The third one is an $$O^*(1.1939^n)$$Oź(1.1939n) time and polynomial (linear) space, which improves over the existing algorithms for exactly solving this problem in general graphs.

Maximal Induced Matchings in Triangle-Free Graphs

An induced matching in a graph is a set of edges whose endpoints induce a 1-regular subgraph. It is known that every n-vertex graph has at most 10n/5≈1.5849n maximal induced matchings, and this bound is the best possible. We prove that every n-vertex triangle-free graph has at most 3n/3≈1.4423n maximal induced matchings; this bound is attained by every disjoint union of copies of the complete bipartite graph K3, 3. Our result implies that all maximal induced matchings in an n-vertex triangle-free graph can be listed in time O(1.4423n), yielding the fastest known algorithm for finding a maximum induced matching in a triangle-free graph.

Moderately exponential time algorithms for the maximum induced matching problem

An induced matching\(M\subseteq E\) in a graph \(G=(V, E)\) is a matching such that no two edges in \(M\) are joined by any third edge of the graph. The Maximum Induced Matching problem is to find an induced matching of maximum cardinality. It is NP-hard. Branch-and-reduce algorithms proposed in the previous results for the Maximum Induced Matching problem use a standard branching rule: for a vertex \(v\), it branches into \(deg(v)+1\) subproblems that either \(v\) is not an endvertex of any edge in \(M\) or \(v\) and one of its neighbor are endvertices of an edge in \(M\). In this paper, we give a simple branch-and-reduce algorithm consisting of a boundary condition, a reduction rule, and a branching rule. Especially, the branching rule only branches the original problem into two subproblems. When the algorithm meets the input instance satisfying the boundary condition, we reduce the Maximum Induced Matching problem to the Maximum Independent Set problem. By using the measure-and-conquer approach to analyze the running time of the algorithm, we show that this algorithm runs in time \(O^{*}(1.4658^n)\) which is faster than previously known algorithms. By adding two branching rules in the simple exact algorithm, we obtain an \(O^{*}(1.4321^n)\)-time algorithm for the Maximum Induced Matching problem. Moreover, we give a moderately exponential time \(\rho \)-approximation algorithm, \(0 < \rho < 1\), for the Maximum Induced Matching problem. For \(\rho =0.5\), the algorithm runs in time \(O^{*}(1.3348^n)\).

Induced Matchings in Subcubic Planar Graphs

Induced Matchings in Subcubic Planar Graphs

Breaking the [formula omitted]-barrier for Irredundance: Two lines of attack

The lower and the upper irredundance numbers of a graph G, denoted ir ( G ) and IR ( G ) , respectively, are conceptually linked to the domination and independence numbers and have numerous relations to other graph parameters. It has been an open question whether determining these numbers for a graph G on n vertices admits exact algorithms running in time faster than the trivial Θ ( 2 n ⋅ poly ( n ) ) enumeration, also called the 2 n -barrier. The main contributions of this article are exact exponential-time algorithms breaking the 2 n -barrier for irredundance. We establish algorithms with running times of O ⁎ ( 1.99914 n ) for computing ir ( G ) and O ⁎ ( 1.9369 n ) for computing IR ( G ) . Both algorithms use polynomial space. The first algorithm uses a parameterized approach to obtain (faster) exact algorithms. The second one is based, in addition, on a reduction to the Maximum Induced Matching problem providing a branch-and-reduce algorithm to solve it.

Approximation hardness of dominating set problems in bounded degree graphs

We study approximation hardness of the Minimum Dominating Set problem and its variants in undirected and directed graphs. Using a similar result obtained by Trevisan for Minimum Set Cover we prove the first explicit approximation lower bounds for various kinds of domination problems (connected, total, independent) in bounded degree graphs. Asymptotically, for degree bound approaching infinity, these bounds almost match the known upper bounds. The results are applied to improve the lower bounds for other related problems such as Maximum Induced Matching and Maximum Leaf Spanning Tree.

Approximability results for the maximum and minimum maximal induced matching problems

An induced matching M of a graph G is a set of pairwise non-adjacent edges such that their end-vertices induce a 1-regular subgraph. An induced matching M is maximal if no other induced matching contains M . The Minimum Maximal Induced Matching problem asks for a minimum maximal induced matching in a given graph. The problem is known to be NP -complete. Here we show that, if P ≠ NP , for any ε > 0 , this problem cannot be approximated within a factor of n 1 − ε in polynomial time, where n denotes the number of vertices in the input graph. The result holds even if the graph in question is restricted to being bipartite. For the related problem of finding an induced matching of maximum size ( Maximum Induced Matching), it is shown that, if P ≠ NP , for any ε > 0 , the problem cannot be approximated within a factor of n 1 / 2 − ε in polynomial time. Moreover, we show that Maximum Induced Matching is NP -complete for planar line graphs of planar bipartite graphs.

Maximum Induced Matchings for Chordal Graphs in Linear Time

The Maximum Induced Matching (MIM) Problem asks for a largest set of pairwise vertex-disjoint edges in a graph which are pairwise of distance at least two. It is well-known that the MIM problem is NP-complete even on particular bipartite graphs and on line graphs. On the other hand, it is solvable in polynomial time for various classes of graphs (such as chordal, weakly chordal, interval, circular-arc graphs and others) since the MIM problem on graph G corresponds to the Maximum Independent Set problem on the square G *=L(G)2 of the line graph L(G) of G, and in some cases, G * is in the same graph class; for example, for chordal graphs G, G * is chordal. The construction of G *, however, requires ${\mathcal{O}}(m^{2})$time, where m is the number of edges in G. Is has been an open problem whether there is a linear-time algorithm for the MIM problem on chordal graphs. We give such an algorithm which is based on perfect elimination order and LexBFS.

A Linear Algorithm for Computing of a Minimum Weight Maximal Induced Matching in an Edge-Weighted Tree

An induced matching M of a graph G is a set of pairwise non-adjacent edges such that their end-vertices induce a 1-regular subgraph. An induced matching M is maximal if no other induced matching contains M. The Minimum Induced Matching problem asks for a minimum maximal induced matching in a given graph. The problem is known to be NP-complete. Here we show that, if G is a tree then this problem can be solved in linear time.