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• W2093844568 abstract "Given a simple graph G, by an L(p,q)-labeling of G we mean a function c that assigns nonnegative integers to its vertices in such a way that if two vertices u, v are adjacent then |c(u)−c(v)|≥p, and if they are at distance 2 then |c(u)−c(v)|≥q. The L(p,q)-labeling problem can be defined as follows: given a graph G and integer t, determine whether there exists an L(p,q)-labeling c of G such that c(V)⊆{0,1,…,t}. In the paper we show that the problem is NP-complete even when restricted to bipartite planar graphs of small maximum degree and for relatively small values of t. More precisely, we prove that: if p<3q then the problem is NP-complete for bipartite planar graphs of maximum degree Δ≤3 and t=p+max{2q,p}; if p=3q then the problem is NP-complete for bipartite planar graphs of maximum degree Δ≤4 and t=6q; if p>3q then the problem is NP-complete for bipartite planar graphs of maximum degree Δ≤4 and t=p+5q. In particular, these results imply that the L(2,1)-labeling problem in planar graphs is NP-complete for t=4, and that the L(p,q)-labeling problem in graphs of maximum degree Δ≤4 is NP-complete for all values of p and q, thus answering two well-known open questions." @default.
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• W2093844568 date "2009-05-01" @default.
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• W2093844568 title "The complexity of the <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML altimg=si14.gif display=inline overflow=scroll><mml:mi>L</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>-labeling problem for bipartite planar graphs of small degree" @default.
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• W2093844568 doi "https://doi.org/10.1016/j.disc.2008.09.028" @default.
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