RECOGNIZING HINGE-FREE LINE GRAPHS AND TOTAL GRAPHS

Abstract

In this paper, we characterize line graphs and total graphs that are hinge-free, i.e., there is no triple of vertices $x,y,z$ such that the distance between $y$ and $z$ increases after $x$ is removed. Based on our characterizations, we show that given a graph $G$ with $n$ vertices and $m$ edges, determining its line graph and total graph to be hinge-free can be solved in O($n+m$) time. Moreover, characterizations of hinge-free iterated line graphs and total graphs are also discussed.