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

Author

Listed:
  • Robert Janczewski

    (Gdańsk University of Technology)

  • Krzysztof Turowski

    (Gdańsk University of Technology)

Abstract

Let $$G=(V,E)$$ G = ( V , E ) be a nonempty graph and $$\xi :E\rightarrow \mathbb {N}$$ ξ : E → N be a function. In the paper we study the computational complexity of the problem of finding vertex colorings $$c$$ c of $$G$$ G such that: (1) $$|c(u)-c(v)|\ge \xi (uv)$$ | c ( u ) - c ( v ) | ≥ ξ ( u v ) for each edge $$uv\in E$$ u v ∈ E ; (2) the edge span of $$c$$ c , i.e. $$\max \{|c(u)-c(v)|:uv\in E\}$$ max { | c ( u ) - c ( v ) | : u v ∈ E } , is minimal. 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 $$G$$ G in $$O(n\log n)$$ O ( n log n ) time, where $$n$$ n is the number of vertices of $$G$$ G .

Suggested Citation

  • Robert Janczewski & Krzysztof Turowski, 2016. "An $$O(n\log n)$$ O ( n log n ) algorithm for finding edge span of cacti," Journal of Combinatorial Optimization, Springer, vol. 31(4), pages 1373-1382, May.
    • Handle: RePEc:spr:jcomop:v:31:y:2016:i:4:d:10.1007_s10878-015-9827-4
    DOI: 10.1007/s10878-015-9827-4
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