Computational complexity of distance edge labeling

Abstract

Abstract. The problem of Distance Edge Labeling is a variant of Distance Vertex Labeling (also known as L2,1 labeling) that has been studied for more than twenty years and has many applications, such as frequency assignment. The Distance Edge Labeling problem asks whether the edges of a given graph can be labeled such that the labels of adjacent edges differ by at least two and the labels of edges at distance two differ by at least one. Labels are chosen from the set {0, 1, . . . , λ} for λ fixed. We present a full classification of its computational complexity-a dichotomy between the polynomially solvable cases and the remaining cases which are NP-complete. We characterise graphs with λ ≤ 4 which leads to a polynomial-time algorithm recognizing the class and we show NPcompleteness for λ ≥ 5 by several reductions from Monotone Not All Equal 3-SAT.

The complexity of L(p,q)-Edge-Labelling

“…from P to NP-complete (indeed, we do not even prove the existence of such a border). The authors of [7] were looking for a more general result, similar to ours, but found the case (p, q) = (2, 1) laborious enough to fill one paper [10]. In fact, their proof settles for us all cases where p ≥ 2q.…”

“…Reduction from Place in article 2 ≤ q /p NAE-3-SAT Section 3.2 1 < q /p ≤ 2 NAE-3-SAT Section 3.1 q /p = 1 5-COL Section 4 ([9]) 2 /3 < q /p ≤ 1 3-COL Section 5 q /p = 2 /3 1-in-3-SAT Section 6 1 /2 < q /p < 2 /3 2-in-4-SAT Section 7 0 < q /p ≤ 1 /2 NAE-3-SAT Appendix ( [7]) Table 1 Table of results and where to find their proofs.…”

“…We follow the exposition of [7], which addresses the L(2,1)-edge-7-labelling problem. With permission we have used (an adaptation) of their diagrams.…”

“…L(1, 1)-Labelling is known as (Proper) Injective Colouring [5,1] and Distance 2 Colouring [11,8] and is studied explicitly in many papers (see [2]). L(2, 1)-Labelling is also studied explicitly in many papers [6,7,4] (see [2]).…”

“…The edge version of the problem, L(p, q)-Edge-k-Labelling, considers an assignment of the labels to the edges instead of the vertices, and now the corresponding distance constraints are placed also on the edges. In [7], the complexity of L(p, q)-Edge-k-Labelling is classified. It is in P for k < 6 and is NP-complete for k ≥ 6.…”