Fan-type theorem for a long path passing through a specified vertex

Let G be a 2-connected graph with maximum degree Δ(G) ≥ d, x and z be distinct vertices of G, and W be a subset of V (G)\{x, z} such that |W| ≤ d - 1. Hirohata proved that if max{d_G(u), d_G(v)} ≥ d for every pair of vertices u, v ∈ V(G)\({x,z} ∪ W) such that d _G(u, v) = 2, then x and z are joined by a path of length at least d - |W|. In this paper, we show that if G satisfies the conditions of Hirohata's theorem, then for any given vertex y such that d_G(y) ≥ d, x and z are joined by a path of length at least d - |W| which contains y.