On Line Graphs Which are Weakly Maximal Clique Irreducible

Abstract

The maximal clique that contains an edge which is not contained in any other maximal cliques is called essential. A graph in which each maximal clique is essential is said to be maximal clique irreducible. Maximal clique irreducible graphs were introduced and studied by W.D. Wallis and G.-H. Zhang in $1990$ $[6]$. We extend the concept and define a graph to be weakly maximal clique irreducible if the set of all essential maximal cliques is a set of least number of maximal cliques that contains every edge. We characterized the graphs for which each induced subgraph is weakly maximal clique irreducible in $[4]$. In this article, we characterize the line graphs which are weakly maximal clique irreducible and also the line graphs which are maximal clique irreducible.