A sufficient condition for [formula omitted]-path graphs being r-connected

Abstract

Given an integer k ⩾ 1 and any graph G , the path graph P k ( G ) has for vertices the paths of length k in G, and two vertices are joined by an edge if and only if the intersection of the corresponding paths forms a path of length k - 1 in G, and their union forms either a cycle or a path of length k + 1 . Path graphs were investigated by Broersma and Hoede [Path graphs, J. Graph Theory 13 (1989), 427–444] as a natural generalization of line graphs. In fact, P 1 ( G ) is the line graph of G . For k = 1 , 2 results on connectivity of P k ( G ) have been given for several authors. In this work, we present a sufficient condition to guarantee that P k ( G ) is connected for k ⩾ 2 if the girth of G is at least ( k + 3 ) / 2 and its minimum degree is at least 4. Furthermore, we determine a lower bound of the vertex-connectivity of P k ( G ) if the girth is at least k + 1 and the minimum degree is at least r + 1 where r ⩾ 2 is an integer.