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Characterization of graphs dominated by induced paths

2007
*
Discrete Mathematics
*

We give

doi:10.1016/j.disc.2005.11.035
fatcat:lflphzl7ovbmdfwdimgthoapke
*a**characterization*, in terms*of*forbidden induced subgraphs,*of*those*graphs*in which every connected induced subgraph has*a**dominating*induced path on at most k vertices (k 3). ... We show, in particular, that k = 4 means precisely the class*of**domination*-*reducible**graphs*, whose original definition applied four types*of*structural reduction. ... The class*of**domination*-*reducible**graphs*is precisely the class*of*hereditarily D 4 -*dominated**graphs*. Corollary 3. ...##
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Characterization The Deletable Set Of Vertices In The ( P - 3 )- Regular Graphs

2011
*
Journal of Mathematics and Computer Science
*

In this paper we

doi:10.22436/jmcs.03.02.07
fatcat:2ce675pxgfg4xpxwvefptoftdq
*characterized*the ( p -3 )-regular*graphs*which have*a*3−deletable and*a*4−deletable set*of*vertices. (2000) : 05C35,05C69, 05C60. Mathematics Subject Classification ...*Characterization*the deletable set*of*vertices In this section we*characterized*the − 3 − regular*graphs*which contain*a*3 − and*a*4 − set*of*vertices. ... Introduction The roots*of*our study in deletable set*of*vertices are in the problem*of**reducibility**of**graphs*. ...##
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Characterizations of minimal dominating sets and the well-dominated property in lexicographic product graphs
[article]

2017
*
arXiv
*
pre-print

Based on this notion, we

arXiv:1701.05886v3
fatcat:hgkaqvgcxfhnzhg7ig34vh5yke
*characterize*the family*of*minimal*dominating*sets in*a*lexicographic product*of*two*graphs*and derive*a**characterization**of*the well-*dominated*lexicographic product*graphs*. ... As*a*side result motivated by this study, we give*a*polynomially testable*characterization**of*well-*dominated**graphs*with*domination*number two, and show, more generally, that well-*dominated**graphs*can ... The main application*of*this notion was*a**characterization**of*the minimal*dominating*sets in nontrivial lexicographic product*graphs*, which led to*a*complete*characterization**of*nontrivial lexicographic ...##
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The Inapproximability of Non NP-hard Optimization Problems
[chapter]

1998
*
Lecture Notes in Computer Science
*

*Dominating*Set: Given

*a*tournament

*graph*G, nd

*a*minimum

*dominating*set for the

*graph*. ... Log k

*Dominating*Set: (k 1) Given

*a*directed

*graph*G = (V; E) in which V = fv 1 ; : : :; v n g and subset fv 1 ; ; v log k n g

*of*V forms

*a*

*dominating*set for G, nd

*a*minimum

*dominating*set for G. ... It

*reduces*any

*graph*G to

*a*hypergraph H such that any vertex cover

*of*size m for H can be used to construct in polynomial time

*a*

*dominating*set

*of*size m for

*graph*G. ...

##
###
Characterization of block graphs with equal 2-domination number and domination number plus one

2007
*
Discussiones Mathematicae Graph Theory
*

Let G be

doi:10.7151/dmgt.1347
fatcat:4witw6cinvbzxesdi75bbastky
*a*simple*graph*, and let p be*a*positive integer.*A*subset is the minimum cardinality among the p-*dominating*sets*of*G. ... If G is*a*nontrivial connected block*graph*, then we show that γ 2 (G) ≥ γ(G) + 1, and we*characterize*all connected block*graphs*with γ 2 (G) = γ(G) + 1. ... In this case we can*reduce*G to*a**graph*G that consists*of*G 0 , the blocks B 1 , B 2 , B 3 and with either*a*leaf to the vertex v or to the vertex x. ...##
###
The inapproximability of non-NP-hard optimization problems

2002
*
Theoretical Computer Science
*

*A*direct connection is established between non-NP-hard problems and

*a*PCP

*characterization*

*of*NP. ...

*A*number

*of*nontrivial approximation-preserving reductions are also presented, making it possible to extend inapproximability results to more natural non-NP-hard problems such as TOURNAMENT

*DOMINATING*... We

*reduce*the PCP

*characterization*

*of*

*a*language L ∈ NP in Corollary 3:6 to MAX SPARSE SAT. ...

##
###
Page 1555 of Mathematical Reviews Vol. , Issue 2002C
[page]

2002
*
Mathematical Reviews
*

In this article, we provide

*a*constructive*characterization**of*those trees with equal independent*domination*and restrained*domination*numbers. ...*A*constructive*characterization**of*those trees with equal independent*domination*and weak*domination*numbers is also obtained.” 2002¢:05053 05C05 05C20 Ziolo, Irmina*A*. ...##
###
Page 2348 of Mathematical Reviews Vol. , Issue 2003d
[page]

2003
*
Mathematical Reviews
*

Given

*a*set*A*CN, we*characterize*cycles C, with sets S C V(C,) that are simultaneously*a*-step*dominating*for precisely those*a*€*A*. ... The first type is formed by*reducing*the num- ber*of*vertices which must be*dominated*by*a*pair*of*vertices and the second by increasing the number*of*steps allowable for dom- ination. ...##
###
Efficient domination in circulant graphs

2013
*
Discrete Mathematics
*

It is proved that if

doi:10.1016/j.disc.2012.12.003
fatcat:sem64ndpibf2hnx3w6faeyzz3q
*a*circulant*graph**of*large degree has an efficient*dominating*set, then either its elements are equally spaced, or the*graph*is the wreath product*of**a*smaller circulant*graph*with ... an efficient*dominating*set and*a*complete*graph*. ... Acknowledgment The research*of*the second author was supported by NSERC. ...
To reflect recent developments in

doi:10.1007/s40840-015-0302-2
fatcat:m7q6g3z7x5gg7kqgaoqk4wn4ve
*graph*theory in*a*timely manner and also to*reduce*the backlog*of*the Bulletin, we decided to edit this special issue on*graph*theory. ... The first contains papers*of*algebraic flavor, starting with*a*paper addressing*graphs**of*ideals*of**a*commutative ring. ... One*of*the deepest topics studied in this area is the Gromov hyperbolicity, and the first paper in this part gives*a**characterization**of*the hyperbolicity*of*periodic*graphs*. ...##
###
Majorizations for the dominant eigenvector of a nonnegative matrix

2008
*
2008 American Control Conference
*

Motivated by network controller design applications, we develop several majorization results for the

doi:10.1109/acc.2008.4586780
dblp:conf/amcc/RoySW08a
fatcat:uc5t4n4qofccncp5fgxytsermi
*dominant*eigenvector*of*an irreducible nonnegative matrix. ... However,*graph*-theoretic*characterization**of*eigenvalues and especially eigenvectors*of*asymmetric matrices is very limited. ...*graph*), and 2) the scaling*of**a*row in the matrix (which corresponds to scaling the "influence"*of**a*node in the*graph*, e.g. by changing*a*controller gain): The result for row-scaling*of*positive matrices ...##
###
Page 4155 of Mathematical Reviews Vol. , Issue 92h
[page]

1992
*
Mathematical Reviews
*

Summary: “

*A**dominating*set S*of**a**graph*G is perfect if each vertex*of*G is*dominated*by exactly one vertex in S. ... We link intersection*graphs*with cographs by introduc- ing ‘component-*reducible*multigraphs’ (multigraph analogues*of*cographs) and relate them to what we call ‘*domination*-*reducible**graphs*’ (also known ...##
###
Page 5188 of Mathematical Reviews Vol. , Issue 96i
[page]

1996
*
Mathematical Reviews
*

*A*set

*of*six

*graph*reduction operations is defined, and it is shown that

*a*

*graph*G € Y iff G can be

*reduced*to an edgeless

*graph*by

*a*finite sequence

*of*these operations. ... (BE-BEL; Minsk) An induced subgraph

*characterization*

*of*

*domination*perfect

*graphs*. (English summary) : J.

*Graph*Theory 20 (1995), no. 3, 375-395. ...

##
###
A witness version of the Cops and Robber game

2009
*
Discrete Mathematics
*

In

doi:10.1016/j.disc.2008.09.032
fatcat:ejvwduuto5er5kmki6eyo4kkgq
*a*special case, this recognition theorem provides*a**characterization*. ... In the case when sightings occur at regular intervals, we present*a*recognition theorem for*graphs*on which*a*single cop suffices to guarantee*a*win. ... Finally, we note that the proof*of*Theorem 9 establishes*a**characterization**of*triangle-free (k, 1)-win*graphs*, namely:*A*triangle-free*graph*G is (k, 1)-win if and only if G can be*reduced*to*a*singleton ...##
###
Minimal Dominating Set Enumeration
[chapter]

2014
*
Encyclopedia of Algorithms
*

Notice that if

doi:10.1007/978-3-642-27848-8_721-1
fatcat:52gg7tmopbaujf4b3femw4odra
*a*class*of**graphs*C has*a*polynomially bounded number*of*minimal separators, then the enumeration*of*minimal connected*dominating*sets can be*reduced*to Dom-Enum [13] . 5. ... For instance it is proved in [11; 13] that every minimal*dominating*set D*of**a*split*graph*G can be*characterized*by D ∩ C(G) where C(G) is the clique*of*G. ...
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