Convex dominating sets in maximal outerplanar graphs

Lemańska, Magdalena; Rivera-Campo, Eduardo; Ziemann, Radosław; Zuazua, Rita; Żyliński, Paweł

doi:10.1016/j.dam.2019.02.029

Cited by 4 publications

(2 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the same paper, the authors conjectured that γ(G) ≤ n 4 for every n-vertex triangulation G, when n is large enough. Since then, several papers have been devoted to either trying to prove that conjecture [23,31,34], or showing thigh bounds, mainly for MOPs, for several variants of the domination number [2,3,4,6,7,9,10,19,26,31,35,37]. In particular, Canales et al [3] proved that γ pr (G) ≤ 2 n 4 for any MOP G of order n ≥ 4, and Henning and Kaemawichanurat [19] showed that γ pr2 (G) ≤ 2 5 n for any MOP G of order n ≥ 5, except for a special family F of MOPs of order 9.…”

Section: Introductionmentioning

confidence: 99%

Paired and semipaired domination in triangulations

Claverol¹,

Hernando²,

Maureso³

et al. 2022

Preprint

View full text Add to dashboard Cite

A dominating set of a graph G is a subset D of vertices such that every vertex not in D is adjacent to at least one vertex in D. A dominating set D is paired if the subgraph induced by its vertices has a perfect matching, and semipaired if every vertex in D is paired with exactly one other vertex in D that is within distance 2 from it. The paired domination number, denoted by γ pr (G), is the minimum cardinality of a paired dominating set of G, and the semipaired domination number, denoted by γ pr2 (G), is the minimum cardinality of a semipaired dominating set of G. A near-triangulation is a biconnected planar graph that admits a plane embedding such that all of its faces are triangles except possibly the outer face. We show in this paper that γ pr (G) ≤ 2 n 4 for any neartriangulation G of order n ≥ 4, and that with some exceptions, γ pr2 (G) ≤ 2n 5 for any near-triangulation G of order n ≥ 5.

show abstract

Section: Introductionmentioning

confidence: 99%

Paired and semipaired domination in triangulations

Claverol¹,

Hernando²,

Maureso³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…In [17], Lemanska et al presented an alternative proof of this last result. The reader is referred to [3,2,4,12,16] for other results in MOPs related to some variants of the domination concept.…”

Section: Introductionmentioning

confidence: 99%

Total domination in plane triangulations

Aguas

García

Peñalver

et al. 2021

Discrete Mathematics

View full text Add to dashboard Cite

A total dominating set of a graph G = (V, E) is a subset D of V such that every vertex in V is adjacent to at least one vertex in D. The total domination number of G, denoted by γ t (G), is the minimum cardinality of a total dominating set of G. A near-triangulation is a biconnected planar graph that admits a plane embedding such that all of its faces are triangles except possibly the outer face. We show in this paper that γ t (G) ≤ 2n 5 for any near-triangulation G of order n ≥ 5, with two exceptions.

show abstract