A parity domination problem in graphs with bounded treewidth and distance-hereditary graphs

Gassner, Elisabeth; Hatzl, Johannes

doi:10.1007/s00607-008-0005-8

Cited by 5 publications

(3 citation statements)

References 20 publications

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“…. We point out that Mu Li designed a polynomial algorithm to derive ML G (x) [27] and some relevant complexity results for determining ML G (x) for tree-like graphs G can be found in [16,32]. For our present purposes, it is pertinent to ask what is the complexity of calculating ML * G (x).…”

Section: Conjecture 8 It Holds For Any Graphmentioning

confidence: 99%

Difference between minimum light numbers of sigma-game and lit-only sigma-game

Wang¹,

Wu²

2009

Preprint

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A configuration of a graph is an assignment of one of two states, on or off, to each vertex of it. A regular move at a vertex changes the states of the neighbors of that vertex. A valid move is a regular move at an on vertex. The following result is proved in this note: given any starting configuration x of a tree, if there is a sequence of regular moves which brings x to another configuration in which there are ℓ on vertices then there must exist a sequence of valid moves which takes x to a configuration with at most ℓ + 2 on vertices. We provide example to show that the upper bound ℓ + 2 is sharp. Some relevant results and conjectures are also reported.

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Section: Conjecture 8 It Holds For Any Graphmentioning

confidence: 99%

Difference between minimum light numbers of sigma-game and lit-only sigma-game

Wang¹,

Wu²

2009

Preprint

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“…We explain these structural properties in more detail in Section 2. This structure has led to the development of a number of algorithms for distance-hereditary graphs [7,29,27,30,41,36,22]. Given the above, we view the vertex deletion problem for distance-hereditary graphs as a first step towards handling Rank-width-t Vertex Deletion.…”

Section: Introductionmentioning

confidence: 99%

A single-exponential fixed-parameter algorithm for distance-hereditary vertex deletion

Eiben

Ganian

Kwon

2018

Journal of Computer and System Sciences

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Vertex deletion problems ask whether it is possible to delete at most k vertices from a graph so that the resulting graph belongs to a specified graph class. Over the past years, the parameterized complexity of vertex deletion to a plethora of graph classes has been systematically researched. Here we present the first single-exponential fixed-parameter tractable algorithm for vertex deletion to distance-hereditary graphs, a well-studied graph class which is particularly important in the context of vertex deletion due to its connection to the graph parameter rank-width. We complement our result with matching asymptotic lower bounds based on the exponential time hypothesis. As an application of our algorithm, we show that a vertex deletion set to distance-hereditary graphs can be used as a parameter which allows single-exponential fixed-parameter tractable algorithms for classical NP-hard problems.

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“…In another recent paper, Gassner and Hatzl [13] discuss an even more general parity domination problem from an algorithmic point of view: for every vertex v, we impose exactly one of the following four constraints:…”

Section: Introductionmentioning

confidence: 99%

Combinatorial properties of a general domination problem with parity constraints

Hatzl

Wagner

2008

Discrete Mathematics

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We consider various properties of a general parity domination problem: given a graph G on n vertices, one is looking for a subset S of the vertex set such that the open/closed neighborhood of each vertex contains an even/odd number of vertices in S (it is prescribed individually for each vertex which of these applies). We define the parameter s(G) to be the number of solvable instances out of 4 n possibilities and study the properties of this parameter. Upper and lower bounds for general graphs and trees are given as well as a remarkable recurrence formula for rooted trees. Furthermore, we give explicit formulas in several special cases and investigate random graphs.

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A parity domination problem in graphs with bounded treewidth and distance-hereditary graphs

Cited by 5 publications

References 20 publications

Difference between minimum light numbers of sigma-game and lit-only sigma-game

Difference between minimum light numbers of sigma-game and lit-only sigma-game

A single-exponential fixed-parameter algorithm for distance-hereditary vertex deletion

Combinatorial properties of a general domination problem with parity constraints

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