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For some β ≥ 1 / 2 , a Δ β -metric graph G = (V , E , w) is a complete edge-weighted graph such that w (v , v) = 0 , w (u , v) = w (v , u) , and w (u , v) ≤ β · (w (u , x) + w (x , v)) for all vertices u , v , x ∈ V . A graph H = (V ′ , E ′) is called a spanning subgraph of G = (V , E) if V ′ = V and E ′ ⊆ E . Given a positive integer p, let H be a spanning subgraph of G satisfying the three conditions: (i) there exists a vertex subset C ⊆ V such that C forms a clique of size p in H; (ii) the set V \ C forms an independent set in H; and (iii) each vertex v ∈ V \ C is adjacent to exactly one vertex in C. The vertices in C are called hubs and the vertices in V \ C are called non-hubs. The Δ β - p -Hub Center Problem ( Δ β - p HCP) is to find a spanning subgraph H of G satisfying all the three conditions such that the diameter of H is minimized. In this paper, we study Δ β - p HCP for all β ≥ 1 2 . We show that for any ϵ > 0 , to approximate Δ β - p HCP to a ratio g (β) - ϵ is NP-hard and we give r (β) -approximation algorithms for the same problem where g (β) and r (β) are functions of β . For 3 - 3 2 < β ≤ 5 + 5 10 , we give an approximation algorithm that reaches the lower bound of approximation ratio g (β) where g (β) = 3 β - 2 β 2 3 (1 - β) if 3 - 3 2 < β ≤ 2 3 and g (β) = β + β 2 if 2 3 ≤ β ≤ 5 + 5 10 . For 5 + 5 10 ≤ β ≤ 1 , we show that g (β) = 4 β 2 + 3 β - 1 5 β - 1 and r (β) = min { β + β 2 , 4 β 2 + 5 β + 1 5 β + 1 } . Additionally, for β ≥ 1 , we show that g (β) = β · 4 β - 1 3 β - 1 and r (β) = min { β 2 + 4 β 3 , 2 β } . For β ≥ 2 , the upper bound on the approximation ratio r (β) = 2 β is linear in β . For 3 - 3 2 < β ≤ 5 + 5 10 , we give an approximation algorithm that reaches the lower bound of approximation ratio g (β) where g (β) = 3 β - 2 β 2 3 (1 - β) if 3 - 3 2 < β ≤ 2 3 and g (β) = β + β 2 if 2 3 ≤ β ≤ 5 + 5 10 . For β ≤ 3 - 3 2 , we show that g (β) = r (β) = 1 , i.e., Δ β - p HCP is polynomial-time solvable. [ABSTRACT FROM AUTHOR]

Abstract: Let be a class of graphs. A graph has -width if there are independent sets in such that can be embedded into a graph such that for every edge in which is not an edge in , there exists an such that both endvertices of are in . For the class of block graphs we show that graphs with -width at most 4 are perfect. We also show that -width is NP-complete and show that it is fixed-parameter tractable. For the class of complete graphs, similar results are also obtained. [Copyright &y& Elsevier]