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NP-hard graph problems and boundary classes of graphs

2007
*
Theoretical Computer Science
*

Any

doi:10.1016/j.tcs.2007.09.013
fatcat:aoqsgkkip5ck7ph7wwmg254uxe
*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. ...##
###
k-tuple domination in graphs

2003
*
Information Processing Letters
*

We also prove that the k-tuple

doi:10.1016/s0020-0190(03)00233-3
fatcat:3a35z4lrxjefpg4ydhowrsbb64
*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*. ...##
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Partitioning the vertex set of a bipartite graph into complete bipartite subgraphs

2014
*
Discrete Mathematics & Theoretical Computer Science
*

It is known that this problem is NP-complete for

doi:10.46298/dmtcs.2090
fatcat:nctfdlgk7vazlogbq3buoqz3gu
*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 ...##
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Exploring the complexity boundary between coloring and list-coloring

2008
*
Annals of Operations Research
*

However, the list-coloring problem is NP-complete for many

doi:10.1007/s10479-008-0391-5
fatcat:u46gfoggonefflcfocepfdwxsu
*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. ...##
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On Balanced Graphs

2005
*
Mathematical programming
*

Characterizations

doi:10.1007/s10107-005-0651-y
fatcat:7go33fqeundtlcuyjzayb36dim
*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. ...##
###
Chordal bipartite graphs of bounded tree- and clique-width

2004
*
Discrete Mathematics
*

A

doi:10.1016/j.disc.2004.02.008
fatcat:vbuf7meexfd6rlzssjfovuka7u
*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. ...##
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Chordal bipartite graphs of bounded tree- and clique-width*1

2004
*
Discrete Mathematics
*

A

doi:10.1016/s0012-365x(04)00156-6
fatcat:tozf2kdjorhs7ji4yxy57we5q4
*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. ...##
###
Algorithmic Aspects of 2-Secure Domination in Graphs
[article]

2020
*
arXiv
*
pre-print

We strengthen the NP-complete result for

arXiv:2002.02408v1
fatcat:xf7uj3ldkve4pm72yrw66dc32a
*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. ...##
###
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. ...

##
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Independent domination in finitely defined classes of graphs

2003
*
Theoretical Computer Science
*

The main result is two su cient conditions for the problem to be NP-hard

doi:10.1016/s0304-3975(02)00586-8
fatcat:ui4x5gajrnaudlendduecqjb4u
*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] ...##
###
Roman Domination in Convex Bipartite Graphs
[article]

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*. ...

##
###
Algorithmic Aspects of Semitotal Domination in Graphs
[article]

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*. ...

##
###
Critical properties of bipartite permutation graphs
[article]

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*. ...

##
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A decidability result for the dominating set problem

2010
*
Theoretical Computer Science
*

We study the following question: given a finite collection

doi:10.1016/j.tcs.2010.08.027
fatcat:t4txbi55efa43ba2usf33rjd5y
*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. ...##
###
Polar SAT and related graphs

2006
*
Journal of Discrete Algorithms
*

As are result, we obtain two narrow hereditary

doi:10.1016/j.jda.2005.01.004
fatcat:xfqwatdrlbdyxmyks2c2q46ohy
*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. ...
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