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A linear-time algorithm for semitotal domination in strongly chordal graphs
[article]

2021
*
arXiv
*
pre-print

*In*this paper, we resolve the complexity of the problem

*in*

*strongly*

*chordal*

*graphs*by designing

*a*

*linear*-

*time*

*algorithm*

*for*the problem. ...

*In*

*a*

*graph*G=(V,E) with no isolated vertex,

*a*

*dominating*set D ⊆ V, is called

*a*

*semitotal*

*dominating*set if

*for*every vertex u ∈ D there is another vertex v ∈ D, such that distance between u and v is at ...

*Algorithm*

*for*

*Semitotal*

*Domination*

*in*

*Strongly*

*Chordal*

*Graphs*

*In*this section, we propose,

*a*

*linear*-

*time*

*algorithm*to compute

*a*minimum semi-TD-set

*in*

*strongly*

*chordal*

*graphs*. ...

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

2017
*
arXiv
*
pre-print

We give

arXiv:1711.10891v1
fatcat:tash2wg3lzekbip73tqaarfeim
*a*polynomial*time**algorithm*to solve the Minimum*Semitotal**Domination*problem*in*interval*graphs*. ...*In*this paper, we show that the*Semitotal**Domination*Decision problem remains NP-complete*for*planar*graphs*, split*graphs*and*chordal*bipartite*graphs*. ... It will be interesting to look*for*the complexity status of the problem*for**strongly**chordal**graphs*, which is*a*subclass of*chordal**graphs*, but superclass of interval*graphs*. ...##
###
Semitotal Domination: New hardness results and a polynomial-time algorithm for graphs of bounded mim-width

2020
*
Theoretical Computer Science
*

*In*particular, we show that it is solvable

*in*polynomial

*time*

*for*the class of

*graphs*of bounded mim-width by

*a*reduction to TOTAL

*DOMINATING*SET and we provide several approximation lower bounds

*for*subclasses ...

*A*

*semitotal*

*dominating*set of

*a*

*graph*G with no isolated vertex is

*a*

*dominating*set D of G such that every vertex

*in*D is within distance two of another vertex

*in*D. ... Acknowledgments We would like to thank the anonymous referees

*for*valuable comments.

*A*. M. would like to thank Martin Milanič and Arne Leitert

*for*useful discussions. ...

##
###
Semitotal Domination: New hardness results and a polynomial-time algorithm for graphs of bounded mim-width
[article]

2018
*
arXiv
*
pre-print

*In*particular, we show that it is solvable

*in*polynomial

*time*

*for*the class of

*graphs*with bounded mim-width by

*a*reduction to Total

*Dominating*Set and we provide several approximation lower bounds

*for*...

*A*

*semitotal*

*dominating*set of

*a*

*graph*G with no isolated vertex is

*a*

*dominating*set D of G such that every vertex

*in*D is within distance two of another vertex

*in*D. ...

*In*view of Lemma 9,

*in*order to provide

*a*polynomial-

*time*

*algorithm*

*for*

*SEMITOTAL*

*DOMINATING*SET restricted to dually

*chordal*

*graphs*, it would then be enough to apply

*a*polynomial-

*time*

*algorithm*

*for*TOTAL ...

##
###
The semitotal domination problem in block graphs

2019
*
Discussiones Mathematicae Graph Theory
*

*In*this paper, we present

*a*

*linear*

*time*

*algorithm*to compute

*a*minimum

*semitotal*

*dominating*set

*in*block

*graphs*. ... The

*semitotal*

*domination*problem is known to be NP-complete

*for*

*chordal*

*graphs*and bipartite

*graphs*as shown

*in*[M.A. Henning and

*A*. ...

*In*this paper, we design

*in*Section 3

*a*

*linear*

*time*

*algorithm*

*for*computing

*a*minimum

*semitotal*

*dominating*set

*in*block

*graphs*,

*a*superclass of trees. ...

##
###
Page 4782 of Mathematical Reviews Vol. , Issue 98H
[page]

1998
*
Mathematical Reviews
*

As an application we get

*a**linear**time*approximation of the diameter*for*weak bipolarizable*graphs*,*a*subclass of HHD-free*graphs*containing all*chordal**graphs*, and an*algorithm*which computes the diameter ... and*a*diametral pair of vertices of*a*distance-hereditary*graph**in**linear**time*.” ...##
###
Bounding the Mim-Width of Hereditary Graph Classes
[article]

2020
*
arXiv
*
pre-print

*A*large number of NP-hard

*graph*problems become polynomial-

*time*solvable on

*graph*classes where the mim-width is bounded and quickly computable. ...

*For*

*a*given

*graph*H, the class of H-free

*graphs*has bounded mim-width if and only if it has bounded clique-width. We show that the same is not true

*for*(H_1,H_2)-free

*graphs*. ... Galby, Munaro and Ries [26] proved that

*Semitotal*

*Dominating*Set is polynomial-

*time*solvable

*for*

*graph*classes where mim-width is bounded and quickly computable. ...

##
###
Reducing the Vertex Cover Number via Edge Contractions
[article]

2022
*
arXiv
*
pre-print

Moreover, unless the ETH fails, the problem does not admit an

arXiv:2202.03322v1
fatcat:6w2aszcbzrf6hbvo3avpwzjmpe
*algorithm*running*in**time*f(k + d) · n^o(k + d)*for*any function f.*In*particular, this answers the open question stated*in*Lima et al. ... CONTRACTION(vc) can be solved*in**time*2^O(d)· n^k - d + O(1). This XP*algorithm*improves the one of Lima et al. ...*domination*number [20] , and the*semitotal**domination*number [18] . ...