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Chordal bipartite, strongly chordal, and strongly chordal bipartite graphs

Terry A. McKee
2003 Discrete Mathematics  
These results motivate a deÿnition of 'strongly chordal bipartite graphs', forming a class intermediate between bipartite interval graphs and chordal bipartite graphs.  ...  Strongly chordal (and chordal bipartite) graphs can be similarly characterized in terms of the distribution of triangles (respectively, quadrilaterals).  ...  chordal bipartite graphs, and of course every complete bipartite graph is strongly chordal bipartite.  ... 
doi:10.1016/s0012-365x(02)00674-x fatcat:fpmoz3jwzjbhrdyupu4xpokkse

Erratum to "Chordal bipartite, strongly chordal, and strongly chordal bipartite graphs"

Terry A. McKee
2003 Discrete Mathematics  
Theorem 5 was not used elsewhere in the paper, except for being erroneously used to motivate the concept of 'strongly chordal bipartite graph' in Section 3.  ...  Theorem 5 of the cited paper asserted that a graph is strongly chordal if and only if every k-cycle C is the sum of k − 2 triangles, each of which contains an edge of C.  ...  Theorem 5 was not used elsewhere in the paper, except for being erroneously used to motivate the concept of 'strongly chordal bipartite graph' in Section 3.  ... 
doi:10.1016/s0012-365x(03)00254-1 fatcat:7cvrbmyzgbdmxo3xobgpr364sq

Strongly chordal and chordal bipartite graphs are sandwich monotone

Pinar Heggernes, Federico Mancini, Charis Papadopoulos, R. Sritharan
2010 Journal of combinatorial optimization  
In this paper we show that strongly chordal graphs and chordal bipartite graphs are sandwich monotone, answering an open question by Bakonyi and Bono from 1997.  ...  In addition, from our results it follows that all strongly chordal graphs and all chordal bipartite graphs with edge constraints can be listed efficiently.  ...  Concluding remarks We have proved that strongly chordal graphs and chordal bipartite graphs are sandwich monotone.  ... 
doi:10.1007/s10878-010-9322-x fatcat:jgltweclezhsndxe5t6hh4g5sa

Strongly Chordal and Chordal Bipartite Graphs Are Sandwich Monotone [chapter]

Pinar Heggernes, Federico Mancini, Charis Papadopoulos, R. Sritharan
2009 Lecture Notes in Computer Science  
bipartite graphs; G and G are both strongly chordal By sandwich monotonicity of strongly chordal ⇒ there is an f ∈ F such that G − f strongly chordal Remove all edges in X ⇒ B − f is chordal bipartite  ...  : unknown minimum chordal bipartite completion: unknown • Strongly chordal graphs and chordal bipartite graphs are sandwich monotone. recognition problem ⇒ O(min{m log n, n 2 }) Paige and Tarjan 1987,  ... 
doi:10.1007/978-3-642-02882-3_40 fatcat:ururdu6ekjg3pgaoguqkvwzqcq

Graph isomorphism completeness for chordal bipartite graphs and strongly chordal graphs

Ryuhei Uehara, Seinosuke Toda, Takayuki Nagoya
2005 Discrete Applied Mathematics  
This paper deals with the graph isomorphism (GI) problem for two graph classes: chordal bipartite graphs and strongly chordal graphs.  ...  It is known that GI problem is GI complete even for some special graph classes including regular graphs, bipartite graphs, chordal graphs, comparability graphs, split graphs, and k-trees with unbounded  ...  Acknowledgements The authors are grateful to Jeremy Spinrad, who informed us that the relative complexity of the GI problem on strongly chordal graphs was still open.  ... 
doi:10.1016/j.dam.2004.06.008 fatcat:fu7cotwjwzgwfiveuzrs5uvr4a

Strongly orderable graphs A common generalization of strongly chordal and chordal bipartite graphs

Feodor F. Dragan
2000 Discrete Applied Mathematics  
This class of graphs, called strongly orderable graphs, generalizes the strongly chordal graphs and the chordal bipartite graphs in a quite natural way.  ...  We consider two characteristic elimination orderings for strongly orderable graphs, one on the vertex set and the second on the edge set, and prove that these graphs can be recognized in O(|V | + |E|)|  ...  Acknowledgements The author is grateful to Andreas Brandst adt and Jeremy Spinrad for many useful discussions.  ... 
doi:10.1016/s0166-218x(99)00149-3 fatcat:lezdqz4b5ffx5pnivdgeoy7upa

On the complexity of the sandwich problems for strongly chordal graphs and chordal bipartite graphs

C.M.H. de Figueiredo, L. Faria, S. Klein, R. Sritharan
2007 Theoretical Computer Science  
Algorithms 19 (1995) 449-473] on graph sandwich problems published in 1995, left the status of sandwich problems for strongly chordal graphs and chordal bipartite graphs open.  ...  We prove that the sandwich problem for strongly chordal graphs is NP-complete. We also give some comments on the computational complexity of the sandwich problem for chordal bipartite graphs.  ...  Acknowledgements We are grateful to the referees for their careful reading and many suggestions, which helped to improve the paper. R.  ... 
doi:10.1016/j.tcs.2007.04.007 fatcat:ijeou6tr5jbytnaqms7eijz3qu

Hamiltonian circuits in chordal bipartite graphs

Haiko Müller
1996 Discrete Mathematics  
As a corollary, HAMILTONIAN CIRCUIT is NP-complete for strongly chordal split graphs.  ...  The main result of this paper is the NP-completeness of the HAMILTONIAN CIRCUIT problem for chordal bipartite graphs. This is proved by a sophisticated reduction from SATISFIABILITY.  ...  In this paper we show the NP-completeness of both problems for chordal bipartite graphs, and as a corollary for strongly chordal split graph.  ... 
doi:10.1016/0012-365x(95)00057-4 fatcat:362pqrhfifgkxjgn3lsmwxh23a

Chordal bipartite completion of colored graphs

R. Sritharan
2008 Discrete Mathematics  
Sritharan, On the complexity of the sandwich problems for strongly chordal graphs and chordal bipartite graphs, Theoret. Comput.  ...  Algorithms 19 (1995) 449-473], in their paper on graph sandwich problems published in 1995, left the status of the sandwich problems for strongly chordal graphs and chordal bipartite graphs open.  ...  Acknowledgments I thank Atif Abueida for his time and careful listening, Luerbio Faria for the figures, and the anonymous referees for their comments.  ... 
doi:10.1016/j.disc.2007.06.004 fatcat:2e5z6ep6jfewplqej4qmbuvcge

Chordal- (k,ℓ)and strongly chordal- (k,ℓ)graph sandwich problems

Fernanda Couto, Luerbio Faria, Sulamita Klein
2014 Journal of the Brazilian Computer Society  
Results: We prove that the STRONGLY CHORDAL-(k, ) GRAPH SANDWICH PROBLEM is NP-complete, for k ≥ 1 and ≥ 1, and that the CHORDAL-(k, ) GRAPH SANDWICH PROBLEM is NP-complete, for positive integers k and  ...  Particurlarly, we are interested in fully classifying the complexity of this problem when we look to the following properties : 'G is a chordal-(k, )-graph' and 'G is a strongly chordal-(k, )-graph', for  ...  Acknowledgements We thank Sritharan for pointing us the result of Dahlhaus and discussing it on Theorem 2 proof. This study is supported by FAPERJ, CNPq and CAPES. Author details  ... 
doi:10.1186/s13173-014-0016-6 fatcat:3vxbp6hlvbbixpa3wgtoo6ut2m

Independence complexes of strongly orderable graphs

Mehmet Akif YETİM
2022 Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics  
In particular, we show that if G is a chordal bipartite graph, then Ind(G) is either contractible or homotopy equivalent to a sphere of dimension at least bp(G) − 1.  ...  We prove that for any finite strongly orderable (generalized strongly chordal) graph G, the independence complex Ind(G) is either contractible or homotopy equivalent to a wedge of spheres of dimension  ...  The class of strongly orderable graphs is firstly introduced by Dahlhaus [5] under the name "generalized strongly chordal graphs" as it constitutes a generalization of strongly chordal graphs and chordal  ... 
doi:10.31801/cfsuasmas.874855 fatcat:vijoc3o6i5fzjmca7tdpqnsy7i

Graph Sandwich Problems

M.C. Golumbic, H. Kaplan, R. Shamir
1995 Journal of Algorithms  
Strongly chordal, chordal bipartite A graph is strongly chordal if it is chordal and contains no sun. A graph is chordal bipartite if it is bipartite and does not contain C n , n ≥ 6.  ...  Recognition problems for strongly chordal and chordal bipartite are polynomially equivalent.  ...  Homogeneous set A homogeneous set H for a graph G(V, E) is a subset of V such that 1 < |H| < |V| and for all v ∈ V \ H, either (v, h) ∈ E for all h ∈ H or (v, h) / ∈ E for all h ∈ H.  ... 
doi:10.1006/jagm.1995.1047 fatcat:tvoyqnusgvcs7hgt43arkkvfuq

The i-chords of cycles and paths

Terry A. McKee
2012 Discussiones Mathematicae Graph Theory  
Results include the following: A graph is strongly chordal if and only if, for i ∈ {4, 6}, every cycle C with |V (C)| ≥ i has an (i/2)-chord.  ...  A graph is a threshold graph if and only if, for i ∈ {4, 5}, every path P with |V (P )| ≥ i has an (i − 2)-chord.  ...  Every ptolemaic graph is strongly chordal [2] (but the strongly chordal graph in Figure 1 is not ptolemaic).  ... 
doi:10.7151/dmgt.1629 fatcat:b4pmg7fi3neonc4ni7yquo6h6i

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

Andreas Brandstädt
2018 Lecture Notes in Computer Science  
ED is solvable in time O(MM) for P 5 -free graphs. chordal Complexity of Efficient Domination: strongly chordal bipartite chordal bipartite convex bip. claw-fr.  ...  EED is polynomial for weakly chordal graphs as well as for permutation graphs and for chordal bipartite graphs.  ...  Corollary.  All edges of any C 4 , and all leaf edges of any paw are excluded.  All mid-edges of diamond, and all external edges of butterfly are forced.  ... 
doi:10.1007/978-3-319-74180-2_1 fatcat:dluwxxtpf5al7jrcyy5smoskky

Treewidth of Chordal Bipartite Graphs

T. Kloks, D. Kratsch
1995 Journal of Algorithms  
Chordal bipartite graph are exactly those bipartite graph in which every cycle of length at least six has a chord.  ...  We present a polynomial time algorithm for the exact computation of the treewidth of all chordal bipartite graphs.  ...  There is a strong connection between chordal bipartite graphs, strongly chordal graphs and totally balanced matrices.  ... 
doi:10.1006/jagm.1995.1037 fatcat:zrqzyj42brdenir6atgc5bw7ha
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