A survey on labeling graphs with a condition at distance two

Yeh, Roger K.

doi:10.1016/j.disc.2005.11.029

Cited by 203 publications

(109 citation statements)

References 39 publications

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“…In this context, L(2, 1)-labeling is generalized into L(p, q)-labeling for arbitrary nonnegative integers p and q, and in fact, we can see that L(1, 0)-labeling (L(p, 0)-labeling, actually) is equivalent to the classical vertex coloring. We can find a lot of related results on L(p, q)-labelings in comprehensive surveys by Calamoneri [11,12] and by Yeh [68]. The survey paper [12] is still updated and we can download the latest version from a web page 1 .…”

Section: (G) a K-l(p Q)-labeling Is An L(p Q)-labeling F : V (G) →mentioning

confidence: 99%

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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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Section: (G) a K-l(p Q)-labeling Is An L(p Q)-labeling F : V (G) →mentioning

confidence: 99%

“…Then, Griggs and Yeh formally introduced the notion of L(p, q)-labeling in [31,67]. Due to its practical importance, the L(2, 1)-labeling problem has been widely studied.…”

Section: (G) a K-l(p Q)-labeling Is An L(p Q)-labeling F : V (G) →mentioning

confidence: 99%

Algorithmic aspects of distance constrained labeling: a survey

Hasunuma

Ishii

Ono

et al. 2014

IJNC

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“…the difference between the largest and the smallest color used. For instance, it was shown that the L( p, q)-labeling problem is NP-complete even for bipartite planar graphs with degree Δ ≤ 4 (Janczewski et al 2009) and polynomially solvable for trees in the case q = 1 (Yeh 2006) and for many simple graph classes: paths, cycles etc. It was also shown that the L(2, 1)-labeling problem is polynomially solvable for cacti (Jonas 1993) and p-almost trees (Fiala and Kratochvil 2001) for every fixed p. Herein we focus on similar minimization criterion: the edge span.…”

Section: Definition 1 Let G = (Vmentioning

confidence: 99%

“…Calamoneri (2006); Yeh (2006) for a survey of results) the authors study the problem of finding colorings with minimal span, i.e. the difference between the largest and the smallest color used.…”

Section: Definition 1 Let G = (Vmentioning

confidence: 99%

An $$O(n\log n)$$ O ( n log n ) algorithm for finding edge span of cacti

Janczewski

Turowski

2015

J Comb Optim

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Let G = (V, E) be a nonempty graph and ξ : E → N be a function. In the paper we study the computational complexity of the problem of finding vertex colorings c of G such that:We show that the problem is NP-hard for subcubic outerplanar graphs of a very simple structure (similar to cycles) and polynomially solvable for cycles and bipartite graphs. Next, we use the last two results to construct an algorithm that solves the problem for a given cactus G in O(n log n) time, where n is the number of vertices of G.

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“…This problem was introduced by Grrigs and Yeh [12,13] in connection with the problem of assigning frequencies in a multihop radio network. Some results of -labelling problem are given below.…”

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confidence: 99%

L(0, 1)-Labelling of Cactus Graphs

Khan¹,

Pal²,

Pal³

2012

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An L(0,1)-labelling of a graph G is an assignment of nonnegative integers to the vertices of G such that the difference between the labels assigned to any two adjacent vertices is at least zero and the difference between the labels assigned to any two vertices which are at distance two is at least one. The span of an L(0,1)-labelling is the maximum label number assigned to any vertex of G. The L(0,1)-labelling number of a graph G, denoted by λ0.1(G) is the least integer k such that G has an L(0,1)-labelling of span k. This labelling has an application to a computer code assignment problem. The task is to assign integer control codes to a network of computer stations with distance restrictions. A cactus graph is a connected graph in which every block is either an edge or a cycle. In this paper, we label the vertices of a cactus graph by L(0,1)-labelling and have shown that, △-1≤λ0.1(G)≤△ for a cactus graph, where △ is the degree of the graph G

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A survey on labeling graphs with a condition at distance two

Abstract: For positive integers, is the smallest k such that there exists a k-L(d 1 , d 2 )-labeling of G. This class of labelings is motivated by the code (or frequency) assignment problem in computer network. This article surveys the results on this labeling problem.

Cited by 203 publications

References 39 publications

Algorithmic aspects of distance constrained labeling: a survey

Algorithmic aspects of distance constrained labeling: a survey

An $$O(n\log n)$$ O ( n log n ) algorithm for finding edge span of cacti

<i>L</i>(0, 1)-Labelling of Cactus Graphs

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A survey on labeling graphs with a condition at distance two

Abstract: For positive integers, is the smallest k such that there exists a k-L(d 1 , d 2 )-labeling of G. This class of labelings is motivated by the code (or frequency) assignment problem in computer network. This article surveys the results on this labeling problem.

Cited by 203 publications

References 39 publications

Algorithmic aspects of distance constrained labeling: a survey

Algorithmic aspects of distance constrained labeling: a survey

An $$O(n\log n)$$ O ( n log n ) algorithm for finding edge span of cacti

&lt;i&gt;L&lt;/i&gt;(0, 1)-Labelling of Cactus Graphs

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<i>L</i>(0, 1)-Labelling of Cactus Graphs