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Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Certifying Algorithms and
Forbidden Induced Subgraphs
P. Heggernes1 D. Kratsch2
1Institutt for InformatikkUniversitetet i Bergen
Norway
2Laboratoire d’Informatique Theorique et AppliqueeUniversite Paul Verlaine - Metz
France
Dagstuhl - GermanyMay 20-25, 2007
1/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Outline
1 Certifying AlgorithmsWhy certifying algorithms ?A Certifying Planarity TestCertificates and authenticationWhat is a Good Certifying Algorithm ?
2 Forbidden Induced SubgraphsCharacterisation of Graph ClassesSubgraphs as CertificatesAuthentication of a Subgraph
3 Previous Work and Our ResultsPrevious WorkOur ResultsCographsTrivially Perfect Graphs
2/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Why certifying algorithms ?A Certifying Planarity TestCertificates and authenticationWhat is a Good Certifying Algorithm ?
Why certifying Algorithms ?
The LEDA Platform of Combinatorial and GeometricComputing : The Planarity Test Story
Software Engineering : hard to avoid bugs in software
Algorithm vs. software : Correctness of an algorithm does notimply that its implementations have no bugs.
Bugs : no termination, wrong result, much too timeconsuming, much too space consuming, etc.
3/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Why certifying algorithms ?A Certifying Planarity TestCertificates and authenticationWhat is a Good Certifying Algorithm ?
What to do about bugs ?
Program Verification : Methods to find and avoid bugs insoftware.
Algorithm Design : Methods to design algorithms such thatbugs in the implementation can be avoided ? ?
Algorithm Design :
Algorithms that support an easy authentication of their resultsusing certificates.The implementation may or may not have bugs.
4/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Why certifying algorithms ?A Certifying Planarity TestCertificates and authenticationWhat is a Good Certifying Algorithm ?
A Planarity Test : Certificates
Linear-time planarity test (e.g. [Hopcroft, Tarjan]) outputs :
YES or NO
planar embedding (certificate) if input planar
Kuratowski graph (certificate) if input non planar
Is this all we need ?
5/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Why certifying algorithms ?A Certifying Planarity TestCertificates and authenticationWhat is a Good Certifying Algorithm ?
A Planarity Test : Certificates
Linear-time planarity test (e.g. [Hopcroft, Tarjan]) outputs :
YES or NO
planar embedding (certificate) if input planar
Kuratowski graph (certificate) if input non planar
Is this all we need ?
5/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Why certifying algorithms ?A Certifying Planarity TestCertificates and authenticationWhat is a Good Certifying Algorithm ?
A Planarity Test : Certificates
Linear-time planarity test (e.g. [Hopcroft, Tarjan]) outputs :
YES or NO
planar embedding (certificate) if input planar
Kuratowski graph (certificate) if input non planar
Is this all we need ?
5/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Why certifying algorithms ?A Certifying Planarity TestCertificates and authenticationWhat is a Good Certifying Algorithm ?
A Planarity Test : Certificates
Linear-time planarity test (e.g. [Hopcroft, Tarjan]) outputs :
YES or NO
planar embedding (certificate) if input planar
Kuratowski graph (certificate) if input non planar
Is this all we need ?
5/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Why certifying algorithms ?A Certifying Planarity TestCertificates and authenticationWhat is a Good Certifying Algorithm ?
A Planarity Test : Authentication
We need to verify the correctness of each certificate ! !
Verify whether the planar embedding is indeed a planarembedding of the input graph.
Verify whether the Kuratowski graph is indeed a subdivision ofthe input graph.
Note that if authentication is positive then the input is certainlyplanar resp. non planar, no matter whether the planarity test hasbugs or not.
6/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Why certifying algorithms ?A Certifying Planarity TestCertificates and authenticationWhat is a Good Certifying Algorithm ?
A Planarity Test : Authentication
We need to verify the correctness of each certificate ! !
Verify whether the planar embedding is indeed a planarembedding of the input graph.
Verify whether the Kuratowski graph is indeed a subdivision ofthe input graph.
Note that if authentication is positive then the input is certainlyplanar resp. non planar, no matter whether the planarity test hasbugs or not.
6/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Why certifying algorithms ?A Certifying Planarity TestCertificates and authenticationWhat is a Good Certifying Algorithm ?
A Planarity Test : Authentication
We need to verify the correctness of each certificate ! !
Verify whether the planar embedding is indeed a planarembedding of the input graph.
Verify whether the Kuratowski graph is indeed a subdivision ofthe input graph.
Note that if authentication is positive then the input is certainlyplanar resp. non planar, no matter whether the planarity test hasbugs or not.
6/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Why certifying algorithms ?A Certifying Planarity TestCertificates and authenticationWhat is a Good Certifying Algorithm ?
A Planarity Test : Authentication
We need to verify the correctness of each certificate ! !
Verify whether the planar embedding is indeed a planarembedding of the input graph.
Verify whether the Kuratowski graph is indeed a subdivision ofthe input graph.
Note that if authentication is positive then the input is certainlyplanar resp. non planar, no matter whether the planarity test hasbugs or not.
6/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Why certifying algorithms ?A Certifying Planarity TestCertificates and authenticationWhat is a Good Certifying Algorithm ?
General framework of a Certifying Recognition Algorithm
Recognition algorithm
input : graph G
output : YES and certificate for membership
output : NO and certificate for non-membership
Authentication
input : graph G , output of recognition algorithmincluding certificate
output : YES if the certificatehas all required properties w.r.t. to the input G
7/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Why certifying algorithms ?A Certifying Planarity TestCertificates and authenticationWhat is a Good Certifying Algorithm ?
General framework of a Certifying Recognition Algorithm
Recognition algorithm
input : graph G
output : YES and certificate for membership
output : NO and certificate for non-membership
Authentication
input : graph G , output of recognition algorithmincluding certificate
output : YES if the certificatehas all required properties w.r.t. to the input G
7/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Why certifying algorithms ?A Certifying Planarity TestCertificates and authenticationWhat is a Good Certifying Algorithm ?
General framework of a Certifying Recognition Algorithm
Recognition algorithm
input : graph G
output : YES and certificate for membership
output : NO and certificate for non-membership
Authentication
input : graph G , output of recognition algorithmincluding certificate
output : YES if the certificatehas all required properties w.r.t. to the input G
7/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Why certifying algorithms ?A Certifying Planarity TestCertificates and authenticationWhat is a Good Certifying Algorithm ?
General framework of a Certifying Recognition Algorithm
Recognition algorithm
input : graph G
output : YES and certificate for membership
output : NO and certificate for non-membership
Authentication
input : graph G , output of recognition algorithmincluding certificate
output : YES if the certificatehas all required properties w.r.t. to the input G
7/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Why certifying algorithms ?A Certifying Planarity TestCertificates and authenticationWhat is a Good Certifying Algorithm ?
General framework of a Certifying Recognition Algorithm
Recognition algorithm
input : graph G
output : YES and certificate for membership
output : NO and certificate for non-membership
Authentication
input : graph G , output of recognition algorithmincluding certificate
output : YES if the certificatehas all required properties w.r.t. to the input G
7/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Why certifying algorithms ?A Certifying Planarity TestCertificates and authenticationWhat is a Good Certifying Algorithm ?
Time and space
Resources needed by a certifying algorithm :
running time of recognition algorithm
running time of authentication algorithm(membership/non-membership)
space needed by certificate
Certifying planarity test :
running time of recognition algorithm : O(n + m)
running time of authentication algorithm for membership :O(n + m)
running time of authentication algorithm fornon-membership : O(n)
8/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Why certifying algorithms ?A Certifying Planarity TestCertificates and authenticationWhat is a Good Certifying Algorithm ?
Time and space
Resources needed by a certifying algorithm :
running time of recognition algorithm
running time of authentication algorithm(membership/non-membership)
space needed by certificate
Certifying planarity test :
running time of recognition algorithm : O(n + m)
running time of authentication algorithm for membership :O(n + m)
running time of authentication algorithm fornon-membership : O(n)
8/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Why certifying algorithms ?A Certifying Planarity TestCertificates and authenticationWhat is a Good Certifying Algorithm ?
Sublinear, Linear and Weak Certificates
sublinear certificates
A certificate is sublinear if the running time of its authenticationalgorithm is tighter than a linear one.
linear certificates
A certificate is linear if the running time of its authenticationalgorithm linear.
weak certificates
A certificate is weak if the running time of its authenticationalgorithm is the same (or even longer) as the one of therecognition algorithm.
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Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Why certifying algorithms ?A Certifying Planarity TestCertificates and authenticationWhat is a Good Certifying Algorithm ?
A Good Certifying Recognition Algorithm
A certifying algorithm to recognize a graph class should preferablyhave the following properties :
Good certifying algorithm
Recognition algorithm has linear running time.
Membership certificates are linear.
Non-membership certificates are sublinear.
Informal and Important :An authentication algorithm should be simple and easy toimplement. It should not redo the computation from scratch and itshould by no means rely on the recognition algorithm.
10/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Characterisation of Graph ClassesSubgraphs as CertificatesAuthentication of a Subgraph
Forbidden Induced Subgraphs
1 Certifying AlgorithmsWhy certifying algorithms ?A Certifying Planarity TestCertificates and authenticationWhat is a Good Certifying Algorithm ?
2 Forbidden Induced SubgraphsCharacterisation of Graph ClassesSubgraphs as CertificatesAuthentication of a Subgraph
3 Previous Work and Our ResultsPrevious WorkOur ResultsCographsTrivially Perfect Graphs
11/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Characterisation of Graph ClassesSubgraphs as CertificatesAuthentication of a Subgraph
Characterisation of Graph Classes
Certifying recognition algorithms for a graph class G often rely oncharacterisations of G.
Characterizations by forbidden (induced) subgraphs are ofparticular interest when designing certifying algorithms :
highly regarded in graph theory
any hereditary graph class can be characterized by theirminimal forbidden induced subgraphs
corresponding certificates often sublinear
corresponding certificates very easy to authenticate
12/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Characterisation of Graph ClassesSubgraphs as CertificatesAuthentication of a Subgraph
Split Graphs, Cographs and Trivially Perfect Graphs
Split Graphs [Foldes & Hammer 77]
A graph is split if and only if it contains no vertex set that induces2K2, C4, or C5.
Cographs
A graph is a cograph if and only if it contains no vertex set thatinduces P4.
Trivially Perfect Graphs [Golumbic 78]
A graph is trivially perfect if and only if it contains no vertexsubset that induces P4 or C4.
13/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Characterisation of Graph ClassesSubgraphs as CertificatesAuthentication of a Subgraph
Subgraphs as Certificates
Small forbidden (induced) subgraphs are ...
natural certificates (typically of non-membership)
sublinear certificates
often not provided by classical recognition algorithms
Convincing certificates
Small forbidden induced subgraphs are particularly convincingcertificates for the user of a corresponding software package. E.g.they can be highlighted in a graphical presentation of the inputgraph.
14/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Characterisation of Graph ClassesSubgraphs as CertificatesAuthentication of a Subgraph
Authentication in O(n) Time
authentication algorithm
The input to the authentication algorithm (to check whetherthe certificate induces a fixed subgraph H of G ) is the inputgraph G = (V ,E ) represented by adjacency lists and acertificate A ⊆ V of constant size.
It can be authenticated in time O(n) whether vertex set Ainduces a subgraph H in the input graph G .
Preferably for some fixed labeling of the vertices of H, therecognition algorithm assigns (by pointers) the vertices of H to theset A, indicating an isomorphism between H and G [A].
15/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Characterisation of Graph ClassesSubgraphs as CertificatesAuthentication of a Subgraph
Authentication in O(n) Time
authentication algorithm
The input to the authentication algorithm (to check whetherthe certificate induces a fixed subgraph H of G ) is the inputgraph G = (V ,E ) represented by adjacency lists and acertificate A ⊆ V of constant size.
It can be authenticated in time O(n) whether vertex set Ainduces a subgraph H in the input graph G .
Preferably for some fixed labeling of the vertices of H, therecognition algorithm assigns (by pointers) the vertices of H to theset A, indicating an isomorphism between H and G [A].
15/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Characterisation of Graph ClassesSubgraphs as CertificatesAuthentication of a Subgraph
Authentication in O(1) Time
authentication in O(1) time
The input to the authentication algorithm (to check whetherthe certificate induces a fixed subgraph H of G ) is the inputgraph G = (V ,E ) represented by ordered adjacency lists anda certificate A ⊆ V of constant size.
It can be authenticated in time O(1) whether vertex set Ainduces a subgraph H of the input graph G .
Use ordered adjacency lists to represent the input graph G .The recognition algorithm passes the set A to theauthentication algorithm by adding pointers to all the edgesand non edges of G [A] in the ordered adjacency lists.
16/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Characterisation of Graph ClassesSubgraphs as CertificatesAuthentication of a Subgraph
Authentication in O(1) Time
authentication in O(1) time
The input to the authentication algorithm (to check whetherthe certificate induces a fixed subgraph H of G ) is the inputgraph G = (V ,E ) represented by ordered adjacency lists anda certificate A ⊆ V of constant size.
It can be authenticated in time O(1) whether vertex set Ainduces a subgraph H of the input graph G .
Use ordered adjacency lists to represent the input graph G .The recognition algorithm passes the set A to theauthentication algorithm by adding pointers to all the edgesand non edges of G [A] in the ordered adjacency lists.
16/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Characterisation of Graph ClassesSubgraphs as CertificatesAuthentication of a Subgraph
Authentication in O(1) Time
authentication in O(1) time
The input to the authentication algorithm (to check whetherthe certificate induces a fixed subgraph H of G ) is the inputgraph G = (V ,E ) represented by ordered adjacency lists anda certificate A ⊆ V of constant size.
It can be authenticated in time O(1) whether vertex set Ainduces a subgraph H of the input graph G .
Use ordered adjacency lists to represent the input graph G .The recognition algorithm passes the set A to theauthentication algorithm by adding pointers to all the edgesand non edges of G [A] in the ordered adjacency lists.
16/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Previous WorkOur ResultsCographsTrivially Perfect Graphs
Previous Work and Our Results
1 Certifying AlgorithmsWhy certifying algorithms ?A Certifying Planarity TestCertificates and authenticationWhat is a Good Certifying Algorithm ?
2 Forbidden Induced SubgraphsCharacterisation of Graph ClassesSubgraphs as CertificatesAuthentication of a Subgraph
3 Previous Work and Our ResultsPrevious WorkOur ResultsCographsTrivially Perfect Graphs
17/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Previous WorkOur ResultsCographsTrivially Perfect Graphs
Previous Work
Linear-time certifying algorithms to recognize ...
planar graphs [LEDA99]
chordal graphs [Tarjan & Yannakakis 84/85]
cographs [Corneil et al. 85]
interval and permutation graphs [Kratsch et al. 06]
proper interval graphs [Hell & Huang 04, Meister 05]
proper interval bigraphs [Hell & Huang 04]
proper circular-arc graphs [Kaplan & Nussbaum 06]
unit circular-arc graphs [Kaplan & Nussbaum 06]
18/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Previous WorkOur ResultsCographsTrivially Perfect Graphs
Our Results I
Linear-time certifying algorithms to recognize ...
split graphs {2K2,C4,C5}-freethreshold graphs {2K2,C4,P4}-freebipartite chain graphs {2K2,C3,C5}-freecobipartite chain graphs {2K2,C3,C5}-freetrivially perfect graphs {C4,P4}-free
19/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Previous WorkOur ResultsCographsTrivially Perfect Graphs
Our Results II
All our certifying algorithms are such that ...
recognition algorithm has linear running time
membership certificate is model of the class
membership certificate is linear
non-membership certificate is a small forbidden inducedsubgraph of the class
non-membership certificate is sublinear
authentication of non-membership certificates in O(1)
20/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Previous WorkOur ResultsCographsTrivially Perfect Graphs
Cographs I
definition by graph operations
Cographs are defined as follows :
A graph consisting of a single vertex is a cograph.
Let G1 and G2 be cographs. Then the join of G1 and G2 isagain a cograph.
Let G1 and G2 be cographs. Then the union of G1 and G2 isagain a cograph.
There are no other cographs.
21/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Previous WorkOur ResultsCographsTrivially Perfect Graphs
Cographs II
P4-free graphs
A graph G is a cograph iff it has no vertex subset that induces aP4.
cotree
A graph G is a cograph iff it has a cotree representation.
The cotree of a cograph is uniquely determined.
22/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Previous WorkOur ResultsCographsTrivially Perfect Graphs
Linear time recognition algorithms
There are various linear-time recognition algorithms for cographs.
[Corneil et al. 85]
linear running time
cotree as membership certificate
vertex set inducing a P4 as non-membership certificate
sublinear non-membership certificate
Does this imply immediately a good certifyingalgorithm ?
23/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Previous WorkOur ResultsCographsTrivially Perfect Graphs
Linear time recognition algorithms
There are various linear-time recognition algorithms for cographs.
[Corneil et al. 85]
linear running time
cotree as membership certificate
vertex set inducing a P4 as non-membership certificate
sublinear non-membership certificate
Does this imply immediately a good certifyingalgorithm ?
23/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Previous WorkOur ResultsCographsTrivially Perfect Graphs
Missing
Find a linear time algorithm that given a graph G = (V ,E ) and atree T , decides whether T is a cotree of G .
EXERCISE
24/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Previous WorkOur ResultsCographsTrivially Perfect Graphs
Missing
Find a linear time algorithm that given a graph G = (V ,E ) and atree T , decides whether T is a cotree of G .
EXERCISE
24/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Previous WorkOur ResultsCographsTrivially Perfect Graphs
Trivially Perfect
Definition
A graph G is trivially perfect if for each induced subgraph H of G ,the number of maximal cliques of H is equal to the maximum sizeof an independent set of H [Golumbic 78].
[Golumbic 78]
A graph is trivially perfect if and only if it contains no vertexsubset that induces P4 or C4.
[Brandstadt et al.]
Trivially perfect graphs are exactly the chordal cographs.
25/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Previous WorkOur ResultsCographsTrivially Perfect Graphs
How to certify ?
Using known recognition algorithms
Both chordal graphs and cographs have linear time certifyingalgorithms [Tarjan & Yannakakis 84/85, Corneil et al. 85,Habib & Paul 05].
Obtaining a forbidden induced subgraph as a certificate ofnon-membership can be done by using those algorithms.
However
The challenge is to give a certificate of membership that can beauthenticated in O(n + m) time.
26/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Previous WorkOur ResultsCographsTrivially Perfect Graphs
Our membership certificates
universal-in-a-component ordering (uco)
A vertex ordering α = (v1, v2, ..., vn) of a graph G is auniversal-in-a-component ordering (uco) if for 1 ≤ i ≤ n, thevertex vi is universal in the connected component ofG [{vi , vi+1, ..., vn}] that vi belongs to.
A graph is trivially perfect if and only if it has a uco.
special type of cotree
A cograph G is a trivially perfect graph if and only if, in the cotreeT of G , every 1-node has at most one child that is a 0-node.
27/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Previous WorkOur ResultsCographsTrivially Perfect Graphs
Linear membership certificates
authentication
For both membership certificates we provide a simpleauthentication algorithm with running time O(n + m).
good certifying algorithm
Thus we obtain two linear time certifying algorithms to recognizetrivially perfect graphs and each has linear membership andsublinear non-membership certificates.
28/32
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Previous WorkOur ResultsCographsTrivially Perfect Graphs
Merci a tous !
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Previous WorkOur ResultsCographsTrivially Perfect Graphs
For Further Reading I
A. Brandstadt, V. B. Le, and J. P. Spinrad,Graph classes : A survey.Philadelphia, SIAM, 1999.
D. G. Corneil, H. Lerchs, and L. Stewart-Burlingham,Complement reducible graphs.Discrete Applied Mathematics 3 :163–174,1981.
D. G. Corneil, Y. Perl, and L. K. Stewart,A linear recognition algorithm for cographs.SIAM J. Comput., 14 :926–934, 1985.
S. Foldes and P. L. Hammer,Split graphs.Congressus Numerantium, 19 :311–315, 1977.
M.C. Golumbic,Trivially perfect graphs.Discrete Math. 24 :105–107, 1978.
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Previous WorkOur ResultsCographsTrivially Perfect Graphs
For Further Reading II
M. C. Golumbic,Algorithmic Graph Theory and Perfect Graphs.Second edition, Annals of Discrete Mathematics 57. Elsevier, 2004.
M. Habib and C. Paul,A simple linear time algorithm for cograph recognition.Discrete Applied Mathematics, 145 :183–197, 2005.
P. Hell and J. Huang,Certifying LexBFS recognition algorithms for proper interval graphs and properinterval bigraphs.SIAM J. Discrete Math., 18 :554–570, 2004.
H. Kaplan and Y. Nussbaum,Certifying algorithms for recognizing proper circular-arc graphs and unitcircular-arc graphs.Proc. of WG 2006 , LNCS 4271, (2006), pp. 289–300.
D. Kratsch, R. M. McConnell, K. Mehlhorn and J. P. Spinrad,Certifying algorithms to recognize interval and permutation graphs.SIAM J. Computing, 36 :326-353, 2006.
Certifying AlgorithmsForbidden Induced Subgraphs
Previous Work and Our Results
Previous WorkOur ResultsCographsTrivially Perfect Graphs
For Further Reading III
K. Mehlhorn and S. Naher,LEDA : A Platform for Combinatorial and Geometric Computing,Cambridge University Press, 1999.
D. Meister,Recognition and computation of minimal triangulations for AT-free claw-free andco-comparability graphs.Discrete Appl. Math., 146 :193–218, 2005.
R. E. Tarjan and M. Yannakakis,Simple linear-time algorithms to test chordality of graphs, test acyclicity ofhypergraphs, and selectively reduce acyclic hypergraphs.SIAM J. Comput., 13 :566–579, 1984.Addendum : SIAM J. Computing, 14 :254–255, 1985.