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### Finding a maximum induced matching in weakly chordal graphs

Kathie Cameron, R. Sritharan, Yingwen Tang
2003 Discrete Mathematics
It is known that ÿnding an induced matching of maximum cardinality in a graph is NP-hard. We show that a maximum induced matching in a weakly chordal graph can be found in polynomial time.  ...  This also demonstrates that the maximum induced matching problem in chordal bipartite graphs can be solved in polynomial time while the problem is known to be NP-hard for bipartite graphs in general.  ...  In this paper, we show that a maximum induced matching in a weakly chordal graph can be found in polynomial time.  ...

### Efficient Domination and Efficient Edge Domination: A Brief Survey [chapter]

Andreas Brandstädt
2018 Lecture Notes in Computer Science
Maximum Induced Matchings Maximum Matching A matching in G is a set of pairwise disjoint edges in G. Maximum Matching (MM) problem: For input graph G, find a matching of maximum size in G. Fact.  ...  Maximum Induced Matching Hence: MIM in polynomial time for  chordal graphs ([B., Hoàng 2005]: in linear time)  weakly chordal graphs, and moreover for  circular-arc graphs  cocomparability graphs  ...  ., Hundt, Nevries LATIN 2010] Let M be a d.i.m. in G. -M contains at least one edge of every odd cycle C 2k+1 and exactly one edge of every C 3 , C 5 , C 7 . -No edge of any C 4 is in M.  ...

### New Min-Max Theorems for Weakly Chordal and Dually Chordal Graphs [chapter]

Arthur H. Busch, Feodor F. Dragan, R. Sritharan
2010 Lecture Notes in Computer Science
A distance-k matching in a graph G is matching M in which the distance between any two edges of M is at least k. A distance-2 matching is more commonly referred to as an induced matching.  ...  In this paper, we show that when G is weakly chordal, the size of the largest induced matching in G is equal to the minimum number of co-chordal subgraphs of G needed to cover the edges of G, and that  ...  Distance-k Matchings in Weakly Chordal Graphs In this section, we observe that the correspondence between maximal cliques of G * and maximal co-chordal subgraphs of G can be adapted to find maximum a distance-k  ...

### Efficient Dominating and Edge Dominating Sets for Graphs and Hypergraphs [chapter]

Andreas Brandstädt, Ragnar Nevries
2016 Encyclopedia of Algorithms
In particular, we solve some open problems and give linear time algorithms for ED and EED on dually chordal graphs.  ...  Let G = (V, E) be a graph. A vertex dominates itself and all its neighbors, i.e., every vertex v ∈ V dominates its closed neighborhood N [v].  ...  The EED problem is solvable in polynomial time for weakly chordal graphs. Proof. In [15] , it is shown that for a weakly chordal graph G, L(G) 2 is weakly chordal.  ...

### Large Induced Subgraphs via Triangulations and CMSO

Fedor V. Fomin, Ioan Todinca, Yngve Villanger
2015 SIAM journal on computing (Print)
For a given graph G = (V, E), the task is to maximize |X| subject to the following: there is a set F ⊆ V such that X ⊆ F , the subgraph G[F ] induced by F is of treewidth at most t, and structure (G[F  ...  Let ϕ be a Counting Monadic Second Order Logic (CMSO) formula and t ≥ 0 be an integer.  ...  For example, Maximum Induced Matching is to find a maximum induced matching which was studied intensively for different graph classes.  ...

### On the strong chromatic index and induced matching of tree-cographs, permutation graphs and chordal bipartite graphs [article]

Ton Kloks, Chin-Ting Ung, Yue-Li Wang
2013 arXiv   pre-print
We show that there exist linear-time algorithms that compute the strong chromatic index and a maximum induced matching of tree-cographs when the decomposition tree is a part of the input.  ...  We also show that there exist efficient algorithms for the strong chromatic index of (bipartite) permutation graphs and of chordal bipartite graphs.  ...  . ⊓ ⊔ Chang proves in [9] that there exists a linear-time algorithm that computes a maximum induced matching in bipartite permutation graphs.  ...

### On greedy matching ordering and greedy matchable graphs [chapter]

Feodor F. Dragan
1997 Lecture Notes in Computer Science
algorithm always yields maximum cardinality matchings for each induced subgraph.  ...  We show that these graphs, called greedy matchable graphs, are a subclass of weakly triangulated graphs and contain strongly chordal graphs and chordal bipartite graphs as proper subclasses.  ...  Moreover, this ordering allows to compute a maximum matching for every induced subgraph of a strongly chordal graph, i.e. to solve a kind of constrained matching problem, when we want to find a maximum  ...

### The complexity of dissociation set problems in graphs

Yury Orlovich, Alexandre Dolgui, Gerd Finke, Valery Gordon, Frank Werner
2011 Discrete Applied Mathematics
Furthermore, the related problem of finding a maximal (by inclusion) dissociation set of minimum size in a given graph is studied, and NP-hardness results for this problem, namely for weakly chordal and  ...  The maximum dissociation set problem, i.e., the problem of finding a dissociation set of maximum size in a given graph is known to be NP-hard for bipartite graphs.  ...  The optimization version of the Maximum Independent Set (the Maximum Induced Matching) problem consists in finding an independent set (an induced matching) of maximum size in a graph G.  ...

### Large induced subgraphs via triangulations and CMSO [article]

Fedor Fomin, Ioan Todinca, Yngve Villanger
2013 arXiv   pre-print
graph, the task is to find a maximum induced subgraph of a given graph containing no graph from Γ as a minor. 3) "Independent Π-packing", where for a fixed finite set of connected graphs Π, the task is  ...  task is to find a maximum induced subgraph of a given graph with at most l vertex-disjoint cycles of length 0 modulo m. 2) "Minimum Γ-deletion", where for a fixed finite set of graphs Γcontaining a planar  ...  For example, Maximum Induced Matching is to find a maximum induced matching which was studied intensively for different graph classes.  ...

### Page 3579 of Mathematical Reviews Vol. , Issue 2004e [page]

2004 Mathematical Reviews
It is known that finding an induced matching of maximum cardinality in a graph is NP-hard. We show that a maximum induced matching in a weakly chordal graph can be found in polynomial time.  ...  in weakly chordal graphs.  ...

### The induced matching and chain subgraph cover problems for convex bipartite graphs

Andreas Brandstädt, Elaine M. Eschen, R. Sritharan
2007 Theoretical Computer Science
We present an O(n 2 )-time algorithm for computing a maximum cardinality induced matching and a minimum cardinality cover by chain subgraphs for convex bipartite graphs.  ...  A minimum chain cover and a maximum induced matching of a given convex bipartite graph can be computed in O(n 2 ) time.  ...  Introduction An induced matching in a graph is a matching that is also an induced subgraph, i.e., no two edges of the matching are joined by an edge in the graph.  ...

### Algorithms for Unipolar and Generalized Split Graphs [article]

Elaine M. Eschen, Xiaoqiang Wang
2013 arXiv   pre-print
We give algorithms on unipolar graphs for finding a maximum independent set and a minimum clique cover in O(n+m) time and for finding a maximum clique and a minimum proper coloring in O(n^2.5/ n), when  ...  A graph G=(V,E) is a unipolar graph if there exits a partition V=V_1 ∪ V_2 such that, V_1 is a clique and V_2 induces the disjoint union of cliques.  ...  Acknowledgement The authors thank John Goldwasser for his suggestion on the analysis of the maximum clique algorithm for unipolar graphs.  ...

### On distance-3 matchings and induced matchings

Andreas Brandstädt, Raffaele Mosca
2011 Discrete Applied Mathematics
We show that, unlike for k = 2, for a chordal graph G, L(G) 3 is not necessarily chordal, and finding a maximum distance-3 matching, and more generally, finding a maximum distance-(2k+1) matching for k  ...  Finding a maximum induced matching is NP-complete even on very restricted bipartite graphs and on claw-free graphs but it can be done efficiently on various classes of graphs such as chordal graphs, based  ...  If we can efficiently find a maximum weight induced matching for the atoms of G, then we can efficiently find a maximum weight induced matching for the entire graph G by using the following recursive approach  ...

### Maximum induced matching problem on hhd-free graphs

Chandra Mohan Krishnamurthy, R. Sritharan
2012 Discrete Applied Mathematics
We show that the maximum induced matching problem can be solved on hhd-free graphs in O(m 2 ) time; hhd-free graphs generalize chordal graphs and the previous best bound was O(m 3 ).  ...  Extending this, we show that for a subclass of hhd-free graphs that is more general than chordal graphs the problem can be solved in linear time.  ...  As G * can be computed in O(m 2 ) time, this yields an O(m 2 ) time algorithm for finding a maximum induced matching in a chordal graph.  ...

### Clique separator decomposition of hole-free and diamond-free graphs and algorithmic consequences

Andreas Brandstädt, Vassilis Giakoumakis, Frédéric Maffray
2012 Discrete Applied Mathematics
This yields efficient algorithms for various optimization problems in this class of graphs.  ...  A paraglider is a graph with five vertices a, b, c, d, e and edges ab, ac, bc, bd, cd, ae, de.  ...  In a complete multipartite graph there is no induced matching of size 2, and in a matched co-bipartite graph there is no induced matching of size 3, so in either class the search for a maximum weight induced  ...
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