Total domination in maximal outerplanar graphs II

We consider (not necessarily proper) colorings of the vertices of a graph where every color is thoroughly dispersed, that is, appears in every open neighborhood. Equivalently, every color is a total dominating set. We define td(G) as the maximum number of colors in such a coloring and FTD(G) as the fractional version thereof. In particular, we show that every claw‐free graph with minimum degree at least two has FTD(G)≥3/2 and this is best possible. For planar graphs, we show that every triangular disc has FTD(G)≥3/2 and this is best possible, and that every planar graph has td(G)≤4 and this is best possible, while we conjecture that every planar triangulation has td(G)≥2. Further, although there are arbitrarily large examples of connected, cubic graphs with td(G)=1, we show that for a connected cubic graph FTD(G)≥2−o(1). We also consider the related concepts in hypergraphs.

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“…In particular, Dorfling et al. showed that, except for two exceptions, every maximal outerplanar graph with order

n \geq 5

has total domination number at most

2 n / 5

. Since a maximal outerplanar graph has minimum degree 2, the most we can hope for is two disjoint total dominating sets (Observation ).…”

Section: Planar and Related Graphsmentioning

confidence: 99%

Thoroughly dispersed colorings

Goddard

Henning

2017

Journal of Graph Theory

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“…Then D ⊆ V (G) is a dominating set if every vertex from V (G)\D is adjacent to some vertex from D. The domination number γ(G) is the minimum cardinality of a dominating set of G. D is a total dominating set if every vertex from V (G) is adjacent to some vertex from D. The total domination number γ t (G) is the minimum cardinality of a total dominating set of G. Note that the total domination number is not defined for graphs that contain isolated vertices, hence unless stated otherwise, all graphs in this paper are isolate-free. For more information on the total domination in graphs see the recent book [17] and papers [11,12].…”

Section: Introductionmentioning

confidence: 99%

On domination-type invariants of Fibonacci cubes and hypercubes

Azarija¹,

Klavžar

Rho³

et al. 2017

Ars Math. Contemp.

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The Fibonacci cube Γ n is the subgraph of the n-dimensional cube Q n induced by the vertices that contain no two consecutive 1s. Using integer linear programming, exact values are obtained for γ t (Γ n), n ≤ 12. Consequently, γ t (Γ n) ≤ 2F n−10 + 21F n−8 holds for n ≥ 11, where F n are the Fibonacci numbers. It is proved that if n ≥ 9, then γ t (Γ n) ≥ (F n+2 − 11)/(n − 3) − 1. Using integer linear programming exact values for the 2packing number, connected domination number, paired domination number, and signed domination number of small Fibonacci cubes and hypercubes are obtained. A conjecture on the total domination number of hypercubes asserting that γ t (Q n) = 2 n−2 holds for n ≥ 6 is also disproved in several ways.

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“…The proofs of these two results in [4,7] are quite different. While Theorem 1 follows from an elegant labeling argument, the proof of Theorem 2 relied on a detailed case analysis; one reason for this difference probably being the existence of the two exceptional graphs.…”

Section: F I G U R Ementioning

confidence: 98%

Dominating sets inducing large components in maximal outerplanar graphs

Alvarado

Dantas

Rautenbach

2017

Journal of Graph Theory

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For a maximal outerplanar graph G of order n at least 3, Matheson and Tarjan showed that G has domination number at most n/3. Similarly, for a maximal outerplanar graph G of order n at least 5, Dorfling, Hattingh, and Jonck showed, by a completely different approach, that G has total domination number at most 2n/5 unless G is isomorphic to one of two exceptional graphs of order 12.We present a unified proof of a common generalization of these two results. For every positive integer k, we specify a set H k of graphs of order at least 4k + 4 and at most 4k 2 − 2k such that every maximal outerplanar graph G of order n at least 2k + 1 that does not belong to H k has a dominating set D of order at most ⌊

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Total domination in maximal outerplanar graphs II

Cited by 30 publications

References 7 publications

Thoroughly dispersed colorings

Thoroughly dispersed colorings

On domination-type invariants of Fibonacci cubes and hypercubes

Dominating sets inducing large components in maximal outerplanar graphs

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