Complexity of Restricted Star Colouring
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Restricted star colouring is a variant of star colouring introduced to design heuristic algorithms to estimate sparse Hessian matrices. For k∈ℕ, a k-restricted star colouring (k-rs colouring) of a graph G is a function f:V(G)→0,1,…,k-1 such that (i)f(x)≠ f(y) for every edge xy of G, and (ii) there is no bicoloured 3-vertex path (P_3) in G with the higher colour on its middle vertex. We show that for k≥ 3, it is NP-complete to test whether a given planar bipartite graph of maximum degree k and arbitrarily large girth admits a k-rs colouring, and thereby answer a problem posed by Shalu and Sandhya (Graphs and Combinatorics, 2016). In addition, it is NP-complete to test whether a 3-star colourable graph admits a 3-rs colouring. We also prove that for all ϵ > 0, the optimization problem of restricted star colouring a 2-degenerate bipartite graph with the minimum number of colours is NP-hard to approximate within n^(1/3)-ϵ. On the positive side, we design (i) a linear-time algorithm to test 3-rs colourability of trees, and (ii) an O(n^3)-time algorithm to test 3-rs colourability of chordal graphs.
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