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NP-hard graph problems and boundary classes of graphs

V.E. Alekseev, R. Boliac, D.V. Korobitsyn, V.V. Lozin
2007 Theoretical Computer Science  
Any graph problem, which is NP-hard in general graphs, becomes polynomial-time solvable when restricted to graphs in special classes. When does a difficult problem become easy?  ...  To answer this question we study the notion of boundary classes.  ...  Acknowledgement Vadim Lozin gratefully acknowledges the support of DIMAP -the Centre for Discrete Mathematics and its Applications at the University of Warwick.  ... 
doi:10.1016/j.tcs.2007.09.013 fatcat:aoqsgkkip5ck7ph7wwmg254uxe

k-tuple domination in graphs

Chung-Shou Liao, Gerard J. Chang
2003 Information Processing Letters  
We also prove that the k-tuple domination problem is NP-complete for split graphs (a subclass of chordal graphs) and for bipartite graphs.  ...  In particular, we give a linear-time algorithm for the k-tuple domination problem in strongly chordal graphs, which is a subclass of chordal graphs and includes trees, block graphs, interval graphs and  ...  NP-completeness results This section establishes NP-complete results for the k-tuple domination problem in split graphs (a subclass of chordal graphs) and in bipartite graphs.  ... 
doi:10.1016/s0020-0190(03)00233-3 fatcat:3a35z4lrxjefpg4ydhowrsbb64

Partitioning the vertex set of a bipartite graph into complete bipartite subgraphs

Oleg Duginov
2014 Discrete Mathematics & Theoretical Computer Science  
It is known that this problem is NP-complete for bipartite graphs. In this paper we investigate the computational complexity of this problem in special subclasses of bipartite graphs.  ...  graphs and bipartite graphs containing no 4-cycles as induced subgraphs.  ...  Partitioning the vertex set of a bipartite graph into bicliques  ... 
doi:10.46298/dmtcs.2090 fatcat:nctfdlgk7vazlogbq3buoqz3gu

Exploring the complexity boundary between coloring and list-coloring

Flavia Bonomo, Guillermo Durán, Javier Marenco
2008 Annals of Operations Research  
However, the list-coloring problem is NP-complete for many subclasses of perfect graphs.  ...  In this work we explore the complexity boundary between vertex coloring and list-coloring on such subclasses of perfect graphs where the former admits polynomial-time algorithms but the latter is NP-complete  ...  Acknowledgements We thank Dominique de Werra, Pavol Hell and Mario Valencia-Pabon for some interesting discussions about different topics on this work.  ... 
doi:10.1007/s10479-008-0391-5 fatcat:u46gfoggonefflcfocepfdwxsu

On Balanced Graphs

Flavia Bonomo, Guillermo Durán, Min Chih Lin, Jayme L Szwarcfiter
2005 Mathematical programming  
Characterizations of balanced graphs by forbidden subgraphs and by clique subgraphs are proved in this work. Using properties of domination we define four subclasses of balanced graphs.  ...  Finally, we analyse the behavior of balanced graphs and these four subclasses under the clique graph operator.  ...  Some subclasses of balanced graphs are characterized by forbidden subgraphs. We can see that in the following two theorems.  ... 
doi:10.1007/s10107-005-0651-y fatcat:7go33fqeundtlcuyjzayb36dim

Chordal bipartite graphs of bounded tree- and clique-width

V. Lozin, D. Rautenbach
2004 Discrete Mathematics  
A bipartite graph is chordal bipartite if every cycle of length at least six has a chord. In the class of chordal bipartite graphs the tree-width and the clique-width are unbounded.  ...  Our main results are that chordal bipartite graphs of bounded vertex degree have bounded tree-width and that k-fork-free chordal bipartite graphs have bounded clique-width, where a k-fork is the graph  ...  This research has been carried out during the visit of Dieter Rautenbach to Rutgers University: assistance of RUTCOR and DIMACS is gratefully acknowledged.  ... 
doi:10.1016/j.disc.2004.02.008 fatcat:vbuf7meexfd6rlzssjfovuka7u

Chordal bipartite graphs of bounded tree- and clique-width*1

V LOZIN
2004 Discrete Mathematics  
A bipartite graph is chordal bipartite if every cycle of length at least six has a chord. In the class of chordal bipartite graphs the tree-width and the clique-width are unbounded.  ...  Our main results are that chordal bipartite graphs of bounded vertex degree have bounded tree-width and that k-fork-free chordal bipartite graphs have bounded clique-width, where a k-fork is the graph  ...  This research has been carried out during the visit of Dieter Rautenbach to Rutgers University: assistance of RUTCOR and DIMACS is gratefully acknowledged.  ... 
doi:10.1016/s0012-365x(04)00156-6 fatcat:tozf2kdjorhs7ji4yxy57we5q4

Algorithmic Aspects of 2-Secure Domination in Graphs [article]

J. Pavan Kumar, P.Venkata Subba Reddy
2020 arXiv   pre-print
We strengthen the NP-complete result for bipartite graphs, by proving this problem is NP-complete for some subclasses of bipartite graphs namely, star convex bipartite, comb convex bipartite graphs.  ...  It is known that 2-SDM is NP-complete for bipartite graphs. In this paper, we prove that the 2-SDM problem is NP-complete for planar graphs and doubly chordal graphs, a subclass of chordal graphs.  ...  Complexity in some subclasses of bipartite graphs To prove the following theorem, we use a restricted version of Exact Cover by 3-Sets, which we denote by RX3C.  ... 
arXiv:2002.02408v1 fatcat:xf7uj3ldkve4pm72yrw66dc32a

Page 3178 of Mathematical Reviews Vol. , Issue 90F [page]

1990 Mathematical Reviews  
In Part I of this series [same journal 55 (1986), 5-14; MR 88d: 05142], the authors discussed three ways of obtaining bipartite graphs from a given graph G, each bipartite graph being unique of its type  ...  Beineke (Fort Wayne, IN) 90f:05117 05C75 Hoang, C. T. (1-RTG-C); Reed, B. A. (1-BELL6) Some classes of perfectly orderable graphs. J. Graph Theory 13 (1989), no. 4, 445-463.  ... 

Independent domination in finitely defined classes of graphs

R. Boliac, V. Lozin
2003 Theoretical Computer Science  
The main result is two su cient conditions for the problem to be NP-hard in a ÿnitely deÿned class of graphs.  ...  We study the independent dominating set problem restricted to graph classes deÿned by ÿnitely many forbidden induced subgraphs.  ...  Several available results deal mainly with subclasses of perfect graphs (such as chordal [14] or co-comparability graphs [20] ) or some other well-structured classes (such as circulararc graphs [7]  ... 
doi:10.1016/s0304-3975(02)00586-8 fatcat:ui4x5gajrnaudlendduecqjb4u

Roman Domination in Convex Bipartite Graphs [article]

Sasmita Rout, Gautam K. Das
2021 arXiv   pre-print
In this paper, we propose a dynamic programming based polynomial time algorithm for RDP in convex bipartite graph.  ...  In the Roman domination problem, an undirected simple graph G(V,E) is given.  ...  In the literature, we observed that some problems are NP-complete for bipartite graphs, but the same problems are solvable in polynomial time in some subclasses of bipartite graphs.  ... 
arXiv:2111.09040v1 fatcat:4ej7xy2hw5a4fpxf26xag32bhm

Algorithmic Aspects of Semitotal Domination in Graphs [article]

Michael A. Henning, Arti Pandey
2017 arXiv   pre-print
In this paper, we show that the Semitotal Domination Decision problem remains NP-complete for planar graphs, split graphs and chordal bipartite graphs.  ...  For a graph G=(V,E), a set D ⊆ V is called a semitotal dominating set of G if D is a dominating set of G, and every vertex in D is within distance 2 of another vertex of D.  ...  One may also look for the complexity status of the problem for subclasses of chordal bipartite graphs, such as bipartite permutation graphs and convex bipartite graphs.  ... 
arXiv:1711.10891v1 fatcat:tash2wg3lzekbip73tqaarfeim

Critical properties of bipartite permutation graphs [article]

Bogdan Alecu, Vadim Lozin, Dmitriy Malyshev
2020 arXiv   pre-print
In the present paper we identify critical subclasses of bipartite permutation graphs of various types.  ...  The class of bipartite permutation graphs enjoys many nice and important properties.  ...  a proper subclass of bipartite permutation graphs.  ... 
arXiv:2010.14467v1 fatcat:4limdlsygvdatfeyksncg3bm2u

A decidability result for the dominating set problem

Vadim V. Lozin
2010 Theoretical Computer Science  
We study the following question: given a finite collection of graphs G 1 , . . . , G k , is the dominating set problem polynomial-time solvable in the class of (G 1 , . . . , G k )-free graphs?  ...  In this paper, we prove the existence of an efficient algorithm that answers this question for k = 2.  ...  Acknowledgements The support of DIMAP (the Centre for Discrete Mathematics and its Applications at the University of Warwick) is gratefully acknowledged.  ... 
doi:10.1016/j.tcs.2010.08.027 fatcat:t4txbi55efa43ba2usf33rjd5y

Polar SAT and related graphs

Igor' Zverovich, Olga I. Zverovich
2006 Journal of Discrete Algorithms  
As are result, we obtain two narrow hereditary subclasses of weakly chordal graphs where Independent Domination is an NP-complete problem.  ...  We introduce two new hereditary classes of graphs, namely polar satgraphs and polar (3, 2)-satgraphs, and we characterize them in terms of forbidden induced subgraphs.  ...  Acknowledgement The authors thank the anonymous referees, whose suggestions helped to improve the presentation of the paper.  ... 
doi:10.1016/j.jda.2005.01.004 fatcat:xfqwatdrlbdyxmyks2c2q46ohy
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