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

Robert Janczewski () and Krzysztof Turowski ()
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Robert Janczewski: Gdańsk University of Technology
Krzysztof Turowski: Gdańsk University of Technology

Journal of Combinatorial Optimization, 2016, vol. 31, issue 4, No 1, 1373-1382

Abstract: 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 .

Keywords: Cacti; Edge span; Vertex coloring; 05C15 (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s10878-015-9827-4

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