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Finding a maximum induced matching in weakly chordal graphs

2003
*
Discrete Mathematics
*

It is known that ÿnding an

doi:10.1016/s0012-365x(02)00803-8
fatcat:rlvvp22rrrh4xoxmlh7zdrs5zu
*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]

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]

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]

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

2015
*
SIAM journal on computing (Print)
*

For

doi:10.1137/140964801
fatcat:zmhblcqudrb5ffkuwmtybcdfye
*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]

2013
*
arXiv
*
pre-print

We show that there exist linear-time algorithms that compute the strong chromatic index and

arXiv:1110.3583v2
fatcat:mmjxyk2gmjdyrc2datxreojoga
*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]

1997
*
Lecture Notes in Computer Science
*

algorithm always yields

doi:10.1007/bfb0024498
fatcat:fuay63zsrfespkb4fqtohswmsy
*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

2011
*
Discrete Applied Mathematics
*

Furthermore, the related problem of

doi:10.1016/j.dam.2011.04.023
fatcat:nb6mvp5p2ffudhlwf5tiyeokdy
*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]

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

2007
*
Theoretical Computer Science
*

We present an O(n 2 )-time algorithm for computing

doi:10.1016/j.tcs.2007.04.006
fatcat:jkmhdfsb5zasdgiiocqhdhzyie
*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]

2013
*
arXiv
*
pre-print

We give algorithms on unipolar

arXiv:1106.6061v2
fatcat:tjjzpfe7tjg2rll4yc7wcs6ixy
*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

2011
*
Discrete Applied Mathematics
*

We show that, unlike for k = 2, for

doi:10.1016/j.dam.2010.05.022
fatcat:a5bfxrhd25hf7evulr26swl6oa
*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

2012
*
Discrete Applied Mathematics
*

We show that the

doi:10.1016/j.dam.2011.08.027
fatcat:scppsdsrwvbmvfuczy2ld3gliy
*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

2012
*
Discrete Applied Mathematics
*

This yields efficient algorithms for various optimization problems

doi:10.1016/j.dam.2011.10.031
fatcat:226muztrbfhyzgchj5i2q3y6xi
*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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