Reconfiguration of List L(2,1)-Labelings in a Graph

Ito, Takehiro; Kawamura, Kazuto; Ono, Hiroyuki; Zhou, Xiao

doi:10.1007/978-3-642-35261-4_7

Cited by 17 publications

(19 citation statements)

References 14 publications

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“…The study of such problems has received considerable attention in recent literature [8,9,13,15,16] and is interesting for a variety of reasons. From an algorithmic standpoint, reconfiguration models dynamic situations in which we seek to transform a solution into a more desirable one, maintaining feasibility during the process.…”

Section: Introductionmentioning

confidence: 99%

“…Some of the problems for which the reconfiguration version has been studied include vertex colouring [1,3,4,6,5], list edge-colouring [14], list L(2,1)-labeling [15], block puzzles [11], independent set [11,13], clique, set cover, integer programming, matching, spanning tree, matroid bases [13], satisfiability [9], shortest path [2,16], subset sum [12], dominating set [10,19], odd cycle transversal, feedback vertex set, and hitting set [19]. For most NP-complete problems, the reconfiguration version has been shown to be PSPACE-complete [13,14,17], while for some problems in P, the reconfiguration question could be either in P [13] or PSPACE-complete [2].…”

Section: Introductionmentioning

confidence: 99%

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Reconfiguration of dominating sets

2015

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Abstract. We explore a reconfiguration version of the dominating set problem, where a dominating set in a graph G is a set S of vertices such that each vertex is either in S or has a neighbour in S. In a reconfiguration problem, the goal is to determine whether there exists a sequence of feasible solutions connecting given feasible solutions s and t such that each pair of consecutive solutions is adjacent according to a specified adjacency relation. Two dominating sets are adjacent if one can be formed from the other by the addition or deletion of a single vertex. For various values of k, we consider properties of D k (G), the graph consisting of a vertex for each dominating set of size at most k and edges specified by the adjacency relation. Addressing an open question posed by Haas and Seyffarth, we demonstrate that D Γ (G)+1 (G) is not necessarily connected, for Γ (G) the maximum cardinality of a minimal dominating set in G. The result holds even when graphs are constrained to be planar, of bounded tree-width, or b-partite for b ≥ 3. Moreover, we construct an infinite family of graphs such that D γ(G)+1 (G) has exponential diameter, for γ(G) the minimum size of a dominating set. On the positive side, we show that Dn−m(G) is connected and of linear diameter for any graph G on n vertices having at least m + 1 independent edges.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Reconfiguration of dominating sets

2015

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“…Thus, it is also natural to consider the list version of L(p, q)-labeling or (p, q)-total labeling. Ito et al [43] studies reassignments of the list version of L(2, 1)-labeling from such a viewpoint.…”

Section: Other Resultsmentioning

confidence: 99%

Algorithmic aspects of distance constrained labeling: a survey

Hasunuma

Ishii

Ono

et al. 2014

IJNC

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Distance constrained labeling problems, e.g., L(p, q)-labeling and (p, q)-total labeling, are originally motivated by the frequency assignment. From the viewpoint of theory, the upper bounds on the labeling numbers and the time complexity of finding a minimum labeling are intensively and extensively studied. In this paper, we survey the distance constrained labeling problems from algorithmic aspects, that is, computational complexity, approximability, exact computation, and so on.

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“…Since then, reconfiguration versions of various problems have been studied, including maximum independent set, minimum vertex cover, maximum matching, shortest path, graph colorability, and many others [8,20,21,22,26]. Typical questions addressed in these works include the structure or the complexity of determining -st-connectivity: whether there is a path from s to t in the reconfiguration graph [8,20,21,22] or -connectivity: whether the reconfiguration graph is connected [5,11,17] or -upper bounds for the diameter of the reconfiguration graph [6,8,21].…”

Section: Background and Motivationmentioning

confidence: 99%

Shortest Reconfiguration Paths in the Solution Space of Boolean Formulas

Mouawad¹,

Nishimura²,

Pathak³

et al. 2017

SIAM J. Discrete Math.

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Abstract. Given a Boolean formula and a satisfying assignment, a flip is an operation that changes the value of a variable in the assignment so that the resulting assignment remains satisfying. We study the problem of computing the shortest sequence of flips (if one exists) that transforms a given satisfying assignment s to another satisfying assignment t of a Boolean formula. Earlier work characterized the complexity of finding any (not necessarily the shortest) sequence of flips from one satisfying assignment to another using Schaefer's framework for classification of Boolean formulas. We build on it to provide a trichotomy for the complexity of finding the shortest sequence of flips and show that it is either in P, NP-complete, or PSPACE-complete. Our result adds to the small set of complexity results known for shortest reconfiguration sequence problems by providing an example where the shortest sequence can be found in polynomial time even though its length is not equal to the symmetric difference of the values of the variables in s and t. This is in contrast to all reconfiguration problems studied so far, where polynomial time algorithms for computing the shortest path were known only for cases where the path modified the symmetric difference only.

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Reconfiguration of List L(2,1)-Labelings in a Graph

Cited by 17 publications

References 14 publications

Reconfiguration of dominating sets

Reconfiguration of dominating sets

Algorithmic aspects of distance constrained labeling: a survey

Shortest Reconfiguration Paths in the Solution Space of Boolean Formulas

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