Deeply asymmetric planar graphs

Aksionov, V. A.; Borodin, O. V.; Mel'nikov, L. S.; Sabidussi, Gert; Stiebitz, Michael; Toft, Bjarne

doi:10.1016/j.jctb.2005.03.002

Cited by 5 publications

(10 citation statements)

References 2 publications

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“…(See [1] for a much stronger bound A(G) ≤ 5 for planar graphs.) In their paper, Erdős and Rényi conjectured that no asymmetric graph attains the upper bound in (8).…”

Section: Erdős and Rényimentioning

confidence: 99%

Paley and the Paley Graphs

Jones¹

2020

Springer Proceedings in Mathematics &Amp; Statistics

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“…(See [1] for a much stronger bound A(G) ≤ 5 for planar graphs.) In their paper, Erdős and Rényi conjectured that no asymmetric graph attains the upper bound in (8).…”

Section: Erdős and Rényimentioning

confidence: 99%

Paley and the Paley Graphs

Jones¹

2020

Springer Proceedings in Mathematics &Amp; Statistics

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“…Suppose that variable x i appears in the clauses c 1 , c 2 , c 3 , c 4 of instance I such that in the induced (embedded) subgraph 4 x i is an anti-clockwise ordering of edges around x i . By looking at G i and considering x i appears positively and negatively, the construction should satisfy one of the following cases:…”

Section: Bipartite Graphsmentioning

confidence: 99%

“…Altogether, this branching has again a branching number of 5, finally leading to an algorithm with running time O * (5 |U| ). Further improvements on this branching should be possible by using [4,Theorem 2], along the lines of thinking elaborated in [2] for a related problem on planar graphs. When we analyze the mentioned algorithm as an exact algorithm, always branching on vertices of smallest degree, which is at least two by our reduction rules, depending on the number n of vertices, branching vectors of (3, 3), (4,4,4), (5,5,5,5), (6,6,6,6,6) (or better) would result, yielding a branching number of 1.32.…”

Section: If We Prioritize the Branching On V /mentioning

confidence: 99%

Extension of Vertex Cover and Independent Set in Some Classes of Graphs

Casel¹,

Fernau²,

Ghadikoalei³

et al. 2019

Lecture Notes in Computer Science

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We consider extension variants of the classical graph problems Vertex Cover and Independent Set. Given a graph G = (V, E) and a vertex set U ⊆ V , it is asked if there exists a minimal vertex cover (resp. maximal independent set) S with U ⊆ S (resp. U ⊇ S). Possibly contradicting intuition, these problems tend to be NP-hard, even in graph classes where the classical problem can be solved in polynomial time. Yet, we exhibit some graph classes where the extension variant remains polynomial-time solvable. We also study the parameterized complexity of theses problems, with parameter |U |, as well as the optimality of simple exact algorithms under the Exponential-Time Hypothesis. All these complexity considerations are also carried out in very restricted scenarios, be it degree or topological restrictions (bipartite, planar or chordal graphs). This also motivates presenting some explicit branching algorithms for degree-bounded instances.We further discuss the price of extension, measuring the distance of U to the closest set that can be extended, which results in natural optimization problems related to extension problems for which we discuss polynomial-time approximability.

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“…We make use of the following structural theorem. [2].) Every connected plane graph with at least two vertices has either 1. two vertices whose degrees sum to at most 5,2.…”

mentioning

confidence: 99%

“…[2].) Every connected plane graph with at least two vertices has either 1. two vertices whose degrees sum to at most 5,2. two vertices at a distance of at most two whose degrees sum to at most 7, 3. a triangular face containing two vertices whose degrees sum to at most 9, or 4. two triangular faces with vertices u and v in common, where the degrees of u and v sum to at most 11.…”

mentioning

confidence: 99%

A bounded search tree algorithm for parameterized face cover

Abu-Khzam

Fernau

Langston

2008

Journal of Discrete Algorithms

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The parameterized complexity of the face cover problem is considered. The input to this problem is a plane graph G with n vertices. The question asked is whether, for a given parameter value k, there exists a set of k or fewer faces whose boundaries collectively cover (contain) every vertex in G. Early attempts achieved run times of O * (12 k ) or worse, by reducing the problem into a special form of dominating set, namely red-blue dominating set, restricted to planar graphs. Here, we consider a direct approach, where some surgical operation is performed on the graph at each branching decision. This paper builds on previous work of the authors and employs a structure theorem of Aksionov et al., with a detailed case analysis, to produce a face cover algorithm that runs in O (4.6056 k + n 2 ) time.We also point to the tight connections with red-blue dominating set on planar graphs via the annotated version of face cover that we consider in our search tree algorithm.Hence, we get both a O (4.6056 k + n 2 ) time algorithm for solving red-blue dominating set on planar graphs and a polynomial time algorithm for producing a linear kernel for annotated face cover.

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Deeply asymmetric planar graphs

Abstract: It is proved that by deleting at most 5 edges every planar (simple) graph of order at least 2 can be reduced to a graph having a non-trivial automorphism. Moreover, the bound of 5 edges cannot be lowered to 4.

Cited by 5 publications

References 2 publications

Paley and the Paley Graphs

Paley and the Paley Graphs

Extension of Vertex Cover and Independent Set in Some Classes of Graphs

A bounded search tree algorithm for parameterized face cover

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