Fixed-parameter complexity of λ-labelings
A λ-labeling of a graph G is an assignment of labels from the set {0,…, λ} to the vertices of G such that vertices at distance of at most two get different labels and adjacent vertices get labels which are at least two apart. We study the minimum value λ= λ( G) such that G admits a λ-labeling. We sh...
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| Published in: | Discrete Applied Mathematics Vol. 113; no. 1; pp. 59 - 72 |
|---|---|
| Main Authors: | , , |
| Format: | Journal Article Conference Proceeding |
| Language: | English |
| Published: |
Lausanne
Elsevier B.V
2001
Amsterdam Elsevier New York, NY |
| Subjects: | |
| ISSN: | 0166-218X, 1872-6771 |
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| Abstract | A
λ-labeling of a graph
G is an assignment of labels from the set {0,…,
λ} to the vertices of
G such that vertices at distance of at most two get different labels and adjacent vertices get labels which are at least two apart. We study the minimum value
λ=
λ(
G) such that
G admits a
λ-labeling. We show that for every fixed value
k⩾4 it is NP-complete to determine whether
λ(
G)⩽
k. We further investigate this problem for sparse graphs (
k-almost trees), extending the already known result for ordinary trees. In a generalization of this problem we wish to find a labeling such that vertices at distance two are assigned labels that differ by at least
q and the labels of adjacent vertices differ by at least
p. We denote the minimum
λ that allows such a labeling by
L(
G;
p,
q). We show several hardness results for
L(
G;
p,
q) including that for any
p>
q⩾1 there is a
λ=
λ(
p,
q) such that deciding if
L(
G;
p,
q)⩽
λ is NP-complete, and that for
p⩾2
q, this decision is NP-complete for every
λ⩾
λ(
p,
q). |
|---|---|
| AbstractList | A
λ-labeling of a graph
G is an assignment of labels from the set {0,…,
λ} to the vertices of
G such that vertices at distance of at most two get different labels and adjacent vertices get labels which are at least two apart. We study the minimum value
λ=
λ(
G) such that
G admits a
λ-labeling. We show that for every fixed value
k⩾4 it is NP-complete to determine whether
λ(
G)⩽
k. We further investigate this problem for sparse graphs (
k-almost trees), extending the already known result for ordinary trees. In a generalization of this problem we wish to find a labeling such that vertices at distance two are assigned labels that differ by at least
q and the labels of adjacent vertices differ by at least
p. We denote the minimum
λ that allows such a labeling by
L(
G;
p,
q). We show several hardness results for
L(
G;
p,
q) including that for any
p>
q⩾1 there is a
λ=
λ(
p,
q) such that deciding if
L(
G;
p,
q)⩽
λ is NP-complete, and that for
p⩾2
q, this decision is NP-complete for every
λ⩾
λ(
p,
q). |
| Author | Kloks, Ton Fiala, Jiřı́ Kratochvı́l, Jan |
| Author_xml | – sequence: 1 givenname: Jiřı́ surname: Fiala fullname: Fiala, Jiřı́ email: fiala@kam.ms.mff.cuni.cz organization: Department of Applied Mathematics and DIMATIA, Charles University, Malostranské nám. 25, 11800 Praha 1, Czech Republic – sequence: 2 givenname: Ton surname: Kloks fullname: Kloks, Ton email: kloks@cs.vu.nl organization: Department of Mathematics and Computer Science, Vrije Universiteit, De Boelelaan 1081 A, 1081 HV Amsterdam, Netherlands – sequence: 3 givenname: Jan surname: Kratochvı́l fullname: Kratochvı́l, Jan email: honza@kam.ms.mff.cuni.cz organization: Department of Applied Mathematics and DIMATIA, Charles University, Malostranské nám. 25, 11800 Praha 1, Czech Republic |
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| Issue | 1 |
| Keywords | Graph cover Channel assignment Graph labeling Fixed-parameter complexity Resource allocation Channel allocation Multigraph NP complete problem Graph theory Workshop Graph labelling Overlay 1999 Complexity |
| Language | English |
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Graph Theory doi: 10.1002/(SICI)1097-0118(199605)22:1<47::AID-JGT7>3.0.CO;2-L – ident: 10.1016/S0166-218X(00)00387-5_BIB2 – start-page: 527 year: 1990 ident: 10.1016/S0166-218X(00)00387-5_BIB17 article-title: Graph algorithms – volume: 47 start-page: 13 year: 1997 ident: 10.1016/S0166-218X(00)00387-5_BIB19 article-title: On distance two labellings of graphs publication-title: Ars Combin. |
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| Snippet | A
λ-labeling of a graph
G is an assignment of labels from the set {0,…,
λ} to the vertices of
G such that vertices at distance of at most two get different... |
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| SubjectTerms | Algorithmics. Computability. Computer arithmetics Applied sciences Channel assignment Computer science; control theory; systems Exact sciences and technology Fixed-parameter complexity Graph cover Graph labeling Theoretical computing |
| Title | Fixed-parameter complexity of λ-labelings |
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