Total Dominating Sets in Maximal Outerplanar Graphs

Lemańska, Magdalena; Zuazua, Rita; Żyliński, Paweł

doi:10.1007/s00373-017-1802-7

Cited by 21 publications

(22 citation statements)

References 7 publications

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“…In this paper, we extend the result proved in [5,17] to the family of neartriangulations and we show that γ t (G) ≤ 2n 5 for any near-triangulation G of order n ≥ 5, apart from the graphs H 1 and H 2 . Thus, we improve the best known upper bound 6 11 n on the total domination number of n-vertex near-triangulations.…”

Section: Introductionsupporting

confidence: 59%

“…In [17], the authors use induction to prove that γ t (T ) ≤ 2n 5 for a MOP T of order n ≥ 21. The two main properties they use are that after contracting a boundary edge of T , the resulting graph is again a MOP, and that there is always a diagonal dividing T into two MOPs, leaving 5, 6, 7 or 8 consecutive boundary edges of C in the smallest one.…”

Section: Near-triangulations and Some Of Their Propertiesmentioning

confidence: 99%

“…In the same paper [18], it is also conjectured that γ(G) ≤ n 4 for any n-vertex triangulation G. King and Pelsmajer proved this conjecture in [15] for triangulations of maximum degree 6, and Plummer et al proved in [20] that if G is an n-vertex Hamiltonian triangulation with minimum degree at least 4, then γ(G) ≤ max{ 2n/7 , 5n/16 }. The upper bound n 3 for triangulations has been recently improved byŠpacapan [22], showing that γ(G) ≤ 17 53 n for any n-vertex triangulation G.…”

Section: Introductionmentioning

confidence: 99%

“…on the total domination number in near-triangulations is proved in Section 4. The proof is based on induction and combines common techniques used when proving results for MOPs, as the ones described in [17], with techniques related to what we call reducible and irreducible near-triangulations, and terminal polygons in irreducible near-triangulations. These concepts are defined in Section 4.…”

mentioning

confidence: 99%

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Total domination in plane triangulations

Aguas

García

Peñalver

et al. 2021

Discrete Mathematics

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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.

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Section: Introductionsupporting

confidence: 59%

Section: Near-triangulations and Some Of Their Propertiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Total domination in plane triangulations

Aguas

García

Peñalver

et al. 2021

Discrete Mathematics

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“…In this section, we prove Theorems 2 and 3. We apply results of O'Rourke [13] in computational geometry that were used in a new proof by Lemańska, Zuazua and Zylinski [10] of an upper bound by Dorfling, Hattingh and Jonck [5] on the size of a total dominating set (a set S of vertices such that each vertex of the graph is adjacent to a vertex in S) of a mop. Before stating these results, we make a related straightforward observation that we will also use.…”

Section: Proofs Of the Upper Boundsmentioning

confidence: 99%

Partial domination of maximal outerplanar graphs

Borg

Kaemawichanurat

2020

Discrete Applied Mathematics

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Several domination results have been obtained for maximal outerplanar graphs (mops). The classical domination problem is to minimize the size of a set S of vertices of an n-vertex graph G such that G − N [S], the graph obtained by deleting the closed neighborhood of S, is null. A classical result of Chvátal is that the minimum size is at most n/3 if G is a mop. Here we consider a modification by allowing G − N [S] to have isolated vertices and isolated edges only. Let ι 1 (G) denote the size of a smallest set S for which this is achieved. We show that if G is a mop on n ≥ 5 vertices, then ι 1 (G) ≤ n/5. We also show that if n 2 is the number of vertices of degree 2, then ι 1 (G) ≤ n+n2 6 if n 2 ≤ n 3 , and ι 1 (G) ≤ n−n2 3 otherwise. We show that these bounds are best possible.

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Isolation of Regular Graphs and k-Chromatic Graphs

Borg

2024

Mediterr. J. Math.

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Given a set $${\mathcal {F}}$$ F of graphs, we call a copy of a graph in $${\mathcal {F}}$$ F an $${\mathcal {F}}$$ F -graph. The $${\mathcal {F}}$$ F -isolation number of a graph G, denoted by $$\iota (G,{\mathcal {F}})$$ ι ( G , F ) , is the size of a smallest set D of vertices of G such that the closed neighborhood of D intersects the vertex sets of the $${\mathcal {F}}$$ F -graphs contained by G (equivalently, $$G - N[D]$$ G - N [ D ] contains no $${\mathcal {F}}$$ F -graph). Thus, $$\iota (G,\{K_1\})$$ ι ( G , { K 1 } ) is the domination number of G. For any integer $$k \ge 1$$ k ≥ 1 , let $${\mathcal {F}}_{1,k}$$ F 1 , k be the set of regular graphs of degree at least $$k-1$$ k - 1 , let $${\mathcal {F}}_{2,k}$$ F 2 , k be the set of graphs whose chromatic number is at least k, and let $${\mathcal {F}}_{3,k}$$ F 3 , k be the union of $${\mathcal {F}}_{1,k}$$ F 1 , k and $${\mathcal {F}}_{2,k}$$ F 2 , k . Thus, k-cliques are members of both $${\mathcal {F}}_{1,k}$$ F 1 , k and $${\mathcal {F}}_{2,k}$$ F 2 , k . We prove that for each $$i \in \{1, 2, 3\}$$ i ∈ { 1 , 2 , 3 } , $$\frac{m+1}{{k \atopwithdelims ()2} + 2}$$ m + 1 k 2 + 2 is a best possible upper bound on $$\iota (G, {\mathcal {F}}_{i,k})$$ ι ( G , F i , k ) for connected m-edge graphs G that are not k-cliques. The bound is attained by infinitely many (non-isomorphic) graphs. The proof of the bound depends on determining the graphs attaining the bound. This appears to be a new feature in the literature on isolation. Among the result’s consequences are a sharp bound of Fenech, Kaemawichanurat, and the present author on the k-clique isolation number and a sharp bound on the cycle isolation number.

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Total Dominating Sets in Maximal Outerplanar Graphs

Cited by 21 publications

References 7 publications

Total domination in plane triangulations

Total domination in plane triangulations

Partial domination of maximal outerplanar graphs

Isolation of Regular Graphs and k-Chromatic Graphs

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