Dominating sets of maximal outerplanar graphs

Tokunaga, Shinichi

doi:10.1016/j.dam.2013.06.025

Cited by 38 publications

(25 citation statements)

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“…Related works Campos and Wakabayashi [2] showed γ 1 (n) = (n + t)/4 , where t is the number of degree-2 nodes (t ≥ 2). This result was independently proved by Tokunaga [11] using a coloring-based and simpler proof.…”

Section: Introductionmentioning

confidence: 88%

γk(n) = max {⌊n/(2k+1)⌋, 1} for Maximal Outerplanar Graphs with n mod (2k+1) ≤ 6

Zhao

2017

Journal of Information Processing

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Let G = (V, E) be an undirected graph with a set V of nodes and a set E of edges, |V| = n. A node v is said to distance-k dominate a node w if w is reachable from v by a path consisting of at most k edges. A set D ⊆ V is said a distance-k dominating set if every node can be distance-k dominated by some v ∈ D. The size of a minimum distance-k dominating set, denoted by γ k (G), is called the distance-k domination number of G. The value γ k (n) is defined by γ k (n) = max{γ k (G) : G has n nodes}. This paper considers γ k (n) for maximal outerplanar graphs. There is a conjecture γ k (n) = max{ n/(2k + 1) , 1}, which was proved for k = 1, 2. This paper gives a unified and simpler proof for k = 1, 2, 3. In fact, a stronger result is shown that for all n > 2k and r = n mod (2k + 1) ≤ 6, there exist at least 2k + 1 − r distinct distance-k dominating sets of size at most n/(2k + 1) , which can be found in linear time.

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Section: Introductionmentioning

confidence: 88%

γk(n) = max {⌊n/(2k+1)⌋, 1} for Maximal Outerplanar Graphs with n mod (2k+1) ≤ 6

Zhao

2017

Journal of Information Processing

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“…T * has n − 3 exterior edges, applying lemma 3.4 it can be 2d-edge covered with ⌊ n 4 ⌋ − 1 edges, and an additional "collapsed edge" at the vertex 3 or 7. This "collapsed edge" also 2d-edge cover the pentagon (3,4,5,6,7). Thus, T can be 2d-edge covered by ⌊ n 4 ⌋ edges.…”

Section: Proofmentioning

confidence: 99%

“…The presence of any of the internal edges (0,7), (0,6), (0,5), (7,1), (7,2) and (7,3) would violate the minimality of m. Thus, the triangle T ′ in T 1 that is bounded by e is (0,4,8). Consider the maximal outerplanar graph T * = T 2 + (0, 4, 5, 6, 7, 8) (see Fig.3(a)).…”

Section: Monitored Bymentioning

confidence: 99%

Monitoring maximal outerplanar graphs

Peñalver

Martins

2014

Electronic Notes in Discrete Mathematics

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“…Also this bound is tight, i. e., i(n) = |_f J. And can also be expressed as a function of the vertices of degree two, following Tokunaga [11], i(n) = | _f+f2J.…”

Section: ) Let G = (V E ) Be a Maximal Outerplanar Graph And S C V mentioning

confidence: 99%

“…In recent years it has received special attention the problem of domination in outerplanar graphs (e.g., [1,2,11]) A graph is outerplanar if it has a crossingfree embedding in the plane such th a t all vertices are on the boundary of its outer face (the unbounded face). An outerplanar graph is maximal if it is not possible to add an edge such th a t the resulting graph is still outerplanar.…”

Section: Introductionmentioning

confidence: 99%