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### Linear-time recognition of bipartite graphs plus two edges

Peter Damaschke
2003 Discrete Mathematics
Cai and Schieber (1997) proved that bipartite graphs plus one edge can be recognized in linear time. We extend their result to bipartite graphs plus two edges.  ...  This gives, as a byproduct, also a simpliÿed solution to the one-edge case. It is a notoriously open question whether recognizing bipartite graphs plus k edges is a ÿxed-parameter tractable problem.  ...  The O(n + m) time recognition of bipartite graphs is folklore [1] . A naive algorithm to recognize bipartite graphs plus k edges would require O((n + m)m k ) time.  ...

### Proper and Planar Drawings of Graphs on Three Layers [chapter]

Matthew Suderman
2006 Lecture Notes in Computer Science
Using the characterization, we then derive corresponding linear-time algorithms for recognizing and drawing proper 3-layer planar graphs.  ...  In this paper, we point out errors in an algorithm of Fößmeier and Kaufmann (CIAC, 1997) for recognizing proper 3-layer planar graphs, and then present a new characterization of this set of graphs that  ...  improving the presentation of this paper.  ...

### Graph classes and forbidden patterns on three vertices [article]

Laurent Feuilloley, Michel Habib
2020 arXiv   pre-print
) that we find, 21 can actually be recognized in linear time.  ...  Damashke started the study of the classes defined by forbidden patterns, a set that contains interval, chordal and bipartite graphs among others.  ...  Acknowledgements: We wish to thank the website graphclasses.org [27] , and its associated book [4] that helped us to navigate through the jungle of graph classes.  ...

### Page 8955 of Mathematical Reviews Vol. , Issue 2003m [page]

2003 Mathematical Reviews
) Linear-time recognition of bipartite graphs plus two edges.  ...  Math. 73 (1997), no. 1, 27-34; MR 97g:05149] proved that bipartite graphs plus one edge can be recognized in linear time. We extend their result to bipartite graphs plus two edges.  ...

### Visibility graphs of towers

Paul Colley, Anna Lubiw, Jeremy Spinrad
1997 Computational geometry
A linear time algorithm is given to recognize the [vertex] visibility graphs of towers, and these graphs are characterized as bipartite permutation graphs with an added Hamiltonian cycle.  ...  A tower is a polygon consisting of two reflex chains sharing one common endpoint, together with one edge joining the other endpoints of the chains.  ...  They present a linear-time recognition algorithm for graphs in the class and polynomial-time algorithms for the Hamilton circuit, crossing number, and minimum fill problems.  ...

### Contents

2003 Discrete Mathematics
Wang and Xuding Zhu Relaxed game chromatic number of graphs 89 P. Damaschke Linear-time recognition of bipartite graphs plus two edges 99 O. Favaron, G.H. Fricke, D. Pritikin and J.  ...  Henning Matching transformation graphs of cubic bipartite plane graphs 27 P. Dankelmann and R.C. Laskar Factor domination and minimum degree 113 T. Dvo\$ r! ak, I. Havel and M.  ...

### Stability number of bull- and chair-free graphs revisited

Andreas Brandstädt, Chı́nh T Hoàng, Van Bang Le
2003 Discrete Applied Mathematics
De Simone showed that prime bull-and chair-free graphs containing a co-diamond are either bipartite or an induced cycle of odd length at least ÿve.  ...  This implies constant-bounded clique width for these graph classes which leads to linear time algorithms for some algorithmic problems.  ...  See [9] for linear time recognition of cographs and [7-9,6] for more informations on P 4 -free graphs.  ...

### Author index to volume

2003 Discrete Mathematics
., Linear-time recognition of bipartite graphs plus two edges (1-3) 99-112 Dankelmann, P. and R.C. Laskar, Factor domination and minimum degree (1-3) 113-119 Du, B., see R.J.\$ r! ak, T., I.  ...  Lee, A MacWilliams-type identity for linear codes on weak order (1-3) 181-194 Koukouvinos, C., see K. Betsumiya (1-3)  ...

### Certifying algorithms for recognizing proper circular-arc graphs and unit circular-arc graphs

Haim Kaplan, Yahav Nussbaum
2009 Discrete Applied Mathematics
We give two new linear-time algorithms, one for recognizing proper circular-arc graphs and the other for recognizing unit circular-arc graphs.  ...  No other previous algorithm for each of these two graph classes provides a certificate for its result.  ...  The characterization of [4] leads to a linear-time recognition algorithm for sparse proper circular-arc graphs.  ...

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

Elaine M. Eschen, Xiaoqiang Wang
2013 arXiv   pre-print
Our algorithm has running time O(nm^'), where m^' is the number of edges in a minimal triangulation of the given graph.  ...  In this paper we present a recognition algorithm for unipolar graphs that utilizes a minimal triangulation of the given graph, and produces a partition when one exists.  ...  Acknowledgement The authors thank John Goldwasser for his suggestion on the analysis of the maximum clique algorithm for unipolar graphs.  ...

### 2-Layer Fan-Planarity: From Caterpillar to Stegosaurus [chapter]

Carla Binucci, Markus Chimani, Walter Didimo, Martin Gronemann, Karsten Klein, Jan Kratochvíl, Fabrizio Montecchiani, Ioannis G. Tollis
2015 Lecture Notes in Computer Science
In a fan-planar drawing of a graph there is no edge that crosses two other independent edges.  ...  We characterize 2-layer fan-planar drawable graphs and describe a linear-time testing and embedding algorithm for biconnected graphs.  ...  Di Giacomo et al. characterized the class of bipartite graphs that admit a RAC drawing on two layers, and described a linear-time testing and embedding algorithm for 2-layer RAC drawable graphs [10] .  ...

### On Satisfiability Problems with a Linear Structure [article]

Serge Gaspers, Christos Papadimitriou, Sigve Hortemo Saether, Jan Arne Telle
2016 arXiv   pre-print
Second, given two linear orders, one for the variables and one for the clauses, we show how to find, in polynomial time, the smallest k such that there is a k-interval bigraph compatible with these two  ...  It was recently shown STV that satisfiability is polynomially solvable when the incidence graph is an interval bipartite graph (an interval graph turned into a bipartite graph by omitting all edges within  ...  Recently another direction for defining linear structure in CSPs was proposed, based in a time honored graph-theoretic conception of linear structure: interval graphs, the intersection graphs of intervals  ...

### Clique-perfectness and balancedness of some graph classes

Flavia Bonomo, Guillermo Durán, Martín D. Safe, Annegret K. Wagler
2014 International Journal of Computer Mathematics
We also give a minimal forbidden induced subgraph characterization and a linear-time recognition algorithm for balancedness of paw-free graphs.  ...  In this work, we give lineartime recognition algorithms and minimal forbidden induced subgraph characterizations of cliqueperfectness and balancedness of P 4 -tidy graphs and a linear-time algorithm for  ...  linear time: in fact, for each component H of G, we can decide in linear time whether or not either H is bipartite or the non-universal vertices of H induce a complete bipartite graph.  ...

### Parallel N-free order recognition

Laurent Viennot
1997 Theoretical Computer Science
Acknowledgements This paper is intentionally structured as the article of Bender et al. [2] .  ...  Parallel recognition algorithms In the following, we develop two parallel algorithms for N-free order recognition. They are both based on Theorem 4 and they compute the bipartite components.  ...  It determines whether the transitive closure of a directed graph is an N-free order in O(n + ml) time where mt is the number of edges of the transitive closure of the input. in this paper, we propose paraiiei  ...

### Certifying Algorithms for Recognizing Interval Graphs and Permutation Graphs

Dieter Kratsch, Ross M. McConnell, Kurt Mehlhorn, Jeremy P. Spinrad
2006 SIAM journal on computing (Print)
We give linear-time certifying algorithms for recognition of interval graphs and permutation graphs.  ...  We show that our certificates of non-membership can be authenticated in O(IV[) time.  ...  A familiar example of a certifying recognition algorithm is a recognition algorithm for bipartite graphs that computes a 2-coloring for bipartite input graphs and an odd cycle for non-bipartite input graphs  ...
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