An Efficient Algorithm for the Detour Hinge Vertex Problem on Trapezoid Graphs

抄録

We consider a simple graph G = (V, E) with vertex set V and edge set E. Let G-u be a subgraph induced by the vertex set V ＼ {u}. The distance δ_G (x, y) is the shortest path's length between the vertices x and y in G. The vertex u ∈ V is a hinge vertex if there exist two vertices x, y ∈ V ＼ {u} such that δ_G-u (x, y) > δ_G (x, y). Let U be a set consisting of all hinge vertices of G. The open neighborhood of u is denoted by N (u). We defined the detour degree of u as det (u) = max{ δ_G-u (x, y) | δ_G-u (x, y) > δ_G (x, y), x, y ∈ N (u)} for u ∈ U. The detour hinge vertex problem aims to determine the hinge vertex u that maximizes det (u) in G. In this study, we propose an efficient algorithm for solving the detour hinge vertex problem on trapezoid graphs that runs in O (n²) time, where n is the number of vertices in the graph.